Algebra of setsFrom Wikipedia, the free encyclopedia

Contents

1 Algebra of sets 11.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The fundamental laws of set algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The principle of duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Some additional laws for unions and intersections . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Some additional laws for complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 The algebra of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 The algebra of relative complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Complement (set theory) 62.1 Relative complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Absolute complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Complements in various programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 List of types of functions 123.1 Relative to set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Relative to an operator (c.q. a group or other structure) . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Relative to a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Relative to an ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Relative to the real/complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Ways of defining functions/Relation to Type Theory . . . . . . . . . . . . . . . . . . . . . . . . . 133.7 Relation to Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Measurable function 154.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

i

ii CONTENTS

4.2 Caveat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Special measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Properties of measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Non-measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Measure (mathematics) 185.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3.1 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.3.2 Measures of infinite unions of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . 205.3.3 Measures of infinite intersections of measurable sets . . . . . . . . . . . . . . . . . . . . . 20

5.4 Sigma-finite measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.7 Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Vitali set 266.1 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Construction and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 1

Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, andcomplementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu-ating expressions, and performing calculations, involving these operations and relations.Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union,the meet operator being intersection, and the complement operator being set complement.

1.1 Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition andmultiplicationare associative and commutative, so are set union and intersection; just as the arithmetic relation less than or equalis reflexive, antisymmetric and transitive, so is the set relation of subset.It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations ofequality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory,and for a full rigorous axiomatic treatment see axiomatic set theory.

1.2 The fundamental laws of set algebra

The binary operations of set union ( ) and intersection ( ) satisfy many identities. Several of these identities orlaws have well established names.

Commutative laws:

A B = B A A B = B A

Associative laws:

(A B) C = A (B C) (A B) C = A (B C)

Distributive laws:

A (B C) = (A B) (A C) A (B C) = (A B) (A C)

The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking.Like addition and multiplication, the operations of union and intersection are commutative and associative, and inter-section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

1

https://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Equality_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Multiplicationhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Commutativityhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relationhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Axiomhttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Binary_operationhttps://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Commutative_operationhttps://en.wikipedia.org/wiki/Associativityhttps://en.wikipedia.org/wiki/Distributivity

2 CHAPTER 1. ALGEBRA OF SETS

Two additional pairs of laws involve the special sets called the empty set and the universal set U ; together withthe complement operator (AC denotes the complement of A). The empty set has no members, and the universal sethas all possible members (in a particular context).

Identity laws:

A = A A U = A

Complement laws:

A AC = U A AC =

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, and U are the identity elements for union and intersection, respectively.Unlike addition and multiplication, union and intersection do not have inverse elements. However the complementlaws give the fundamental properties of the somewhat inverse-like unary operation of set complementation.The preceding five pairs of lawsthe commutative, associative, distributive, identity and complement lawsencompassall of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them.Note that if the complement laws are weakened to the rule (AC)C = A , then this is exactly the algebra of proposi-tional linear logic.

1.3 The principle of duality

See also: Duality (order theory)

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other byinterchanging and , and also and U.These are examples of an extremely important and powerful property of set algebra, namely, the principle of dualityfor sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions andintersections, interchanging U and and reversing inclusions is also true. A statement is said to be self-dual if it isequal to its own dual.

1.4 Some additional laws for unions and intersections

The following proposition states six more important laws of set algebra, involving unions and intersections.PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:

idempotent laws:

A A = A A A = A

domination laws:

A U = U A =

absorption laws:

A (A B) = A A (A B) = A

https://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Universal_sethttps://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Identity_elementhttps://en.wikipedia.org/wiki/Inverse_elementhttps://en.wikipedia.org/wiki/Unary_operationhttps://en.wikipedia.org/wiki/Linear_logichttps://en.wikipedia.org/wiki/Duality_(order_theory)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Idempotenthttps://en.wikipedia.org/wiki/Absorption_law

1.5. SOME ADDITIONAL LAWS FOR COMPLEMENTS 3

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws statedabove. As an illustration, a proof is given below for the idempotent law for union.Proof:

The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law forunion, namely the idempotent law for intersection.Proof:

Intersection can be expressed in terms of set difference :A B = A (AB)

1.5 Some additional laws for complements

The following proposition states five more important laws of set algebra, involving complements.PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgans laws:

(A B)C = AC BC

(A B)C = AC BC

double complement or Involution law:

(AC)C = A

complement laws for the universal set and the empty set:

C = U UC =

Notice that the double complement law is self-dual.The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies thecomplement laws. In other words, complementation is characterized by the complement laws.PROPOSITION 5: Let A and B be subsets of a universe U, then:

uniqueness of complements:

If A B = U , and A B = , then B = AC

1.6 The algebra of inclusion

The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is apartial order.PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:

A A

antisymmetry:

A B and B A if and only if A = B

transitivity:

https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/De_Morgan%2527s_lawshttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Binary_relationhttps://en.wikipedia.org/wiki/Partial_orderhttps://en.wikipedia.org/wiki/Reflexive_relationhttps://en.wikipedia.org/wiki/Antisymmetric_relationhttps://en.wikipedia.org/wiki/Transitive_relation

4 CHAPTER 1. ALGEBRA OF SETS

If A B and B C , then A C

The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, andhence together with the distributive and complement laws above, show that it is a Boolean algebra.PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element:

A S

existence of joins:

A A B If A C and B C , then A B C

existence of meets:

A B A If C A and C B , then C A B

The following proposition says that the statement A B is equivalent to various other statements involving unions,intersections and complements.PROPOSITION 8: For any two sets A and B, the following are equivalent:

A B A B = A A B = B AB = BC AC

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of setunion or set intersection, which means that the notion of set inclusion is axiomatically superfluous.

1.7 The algebra of relative complements

The following proposition lists several identities concerning relative complements and set-theoretic differences.PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

C \ (A B) = (C \A) (C \B) C \ (A B) = (C \A) (C \B) C \ (B \A) = (A C) (C \B) (B \A) C = (B C) \A = B (C \A) (B \A) C = (B C) \ (A \ C) A \A = \A = A \ = A B \A = AC B (B \A)C = A BC

U \A = AC

A \ U =

https://en.wikipedia.org/wiki/Power_sethttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Boolean_algebra_(structure)https://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Greatest_elementhttps://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Lattice_(order)https://en.wikipedia.org/wiki/Complement_(set_theory)

1.8. SEE ALSO 5

1.8 See also -algebra is an algebra of sets, completed to include countably infinite operations.

Axiomatic set theory

Field of sets

Naive set theory

Set (mathematics)

1.9 References Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. TheAlgebra of Sets, pp 1623

Courant, Richard, Herbert Robbins, Ian Stewart,What is mathematics?: An Elementary Approach to Ideas andMethods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. SUPPLEMENT TO CHAPTER IITHE ALGEBRA OF SETS

1.10 External links Operations on Sets at ProvenMath

https://en.wikipedia.org/wiki/%CE%A3-algebrahttps://en.wikipedia.org/wiki/Axiomatic_set_theoryhttps://en.wikipedia.org/wiki/Field_of_setshttps://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Special:BookSources/0486638294http://books.google.com/books?id=3-nrPB7BQKMC&pg=PA16#v=onepage&q&f=falsehttp://books.google.com/books?id=3-nrPB7BQKMC&pg=PA16#v=onepage&q&f=falsehttps://en.wikipedia.org/wiki/Special:BookSources/9780195105193http://books.google.com/books?id=UfdossHPlkgC&pg=PA17-IA8&dq=%2522algebra+of+sets%2522&hl=en&ei=k8-RTdXoF4K2tgfM-p1v&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDYQ6AEwAg#v=onepage&q=%2522algebra%2520of%2520sets%2522&f=falsehttp://books.google.com/books?id=UfdossHPlkgC&pg=PA17-IA8&dq=%2522algebra+of+sets%2522&hl=en&ei=k8-RTdXoF4K2tgfM-p1v&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDYQ6AEwAg#v=onepage&q=%2522algebra%2520of%2520sets%2522&f=falsehttp://www.apronus.com/provenmath/btheorems.htm

Chapter 2

Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complementof A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are consideredto be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.

2.1 Relative complement

If A and B are sets, then the relative complement of A in B,[1] also termed the set-theoretic difference of B andA,[2] is the set of elements in B, but not in A.

The relative complement of A (left circle) in B (right circle): B Ac = B \A

The relative complement of A in B is denoted B A according to the ISO 31-11 standard (sometimes written B A,but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b a, where b is taken fromB and a from A).

6

https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/ISO_31-11#Sets

2.2. ABSOLUTE COMPLEMENT 7

Formally

B \A = {x B |x / A}.

Examples:

{1,2,3} {2,3,4} = {1} {2,3,4} {1,2,3} = {4} If R is the set of real numbers and Q is the set of rational numbers, then R \ Q = I is the set ofirrational numbers.

The following lists some notable properties of relative complements in relation to the set-theoretic operations of unionand intersection.If A, B, and C are sets, then the following identities hold:

C (A B) = (C A)(C B) C (A B) = (C A)(C B) C (B A) = (C A)(C B)

[ Alternately written: A (B C) = (A B)(A C) ]

(B A) C = (B C) A = B(C A) (B A) C = (B C) (A C) A A = A = A = A

2.2 Absolute complement

If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simplycomplement) of A, and is denoted by Ac or sometimes A. The same set often[3] is denoted by UA or A if U isfixed, that is:

Ac = U A.

For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of evennumbers.The following lists some important properties of absolute complements in relation to the set-theoretic operations ofunion and intersection.If A and B are subsets of a universe U, then the following identities hold:

De Morgans laws:[1]

(A B)c = Ac Bc. (A B)c = Ac Bc.

Complement laws:[1]

A Ac = U. A Ac = . c = U.

https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Irrational_numberhttps://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Universe_(mathematics)https://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Operation_(mathematics)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Universe_(mathematics)https://en.wikipedia.org/wiki/De_Morgan%2527s_laws

8 CHAPTER 2. COMPLEMENT (SET THEORY)

The absolute complement of A in U : Ac = U \A

U c = . IfA B then ,Bc Ac.

(this follows from the equivalence of a conditional with its contrapositive)

Involution or double complement law:

(Ac)c = A.

Relationships between relative and absolute complements:

A B = A Bc (A B)c = Ac B

The first two complement laws above shows that if A is a non-empty, proper subset of U, then {A, Ac} is a partitionof U.

2.3 Notation

In the LaTeX typesetting language, the command \setminus[4] is usually used for rendering a set difference symbol,which is similar to a backslash symbol. When rendered the \setminus command looks identical to \backslash exceptthat it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. Avariant \smallsetminus is available in the amssymb package.

2.4 Complements in various programming languages

Some programming languages allow for manipulation of sets as data structures, using these operators or functions toconstruct the difference of sets a and b:

https://en.wikipedia.org/wiki/Contrapositivehttps://en.wikipedia.org/wiki/Involution_(mathematics)https://en.wikipedia.org/wiki/Proper_subsethttps://en.wikipedia.org/wiki/Partition_of_a_sethttps://en.wikipedia.org/wiki/LaTeXhttps://en.wikipedia.org/wiki/Backslashhttps://en.wikipedia.org/wiki/Set_(computer_science)

2.4. COMPLEMENTS IN VARIOUS PROGRAMMING LANGUAGES 9

.NET Framework a.Except(b);

C++ set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());

Clojure (clojure.set/difference a b)[5]

Common Lisp set-difference, nset-difference[6]

F# Set.difference a b[7]

or

a - b[8]

Falcon diff = a - b[9]

Haskell difference a b

a \\ b[10]

Java diff = a.clone();

diff.removeAll(b);[11]

Julia setdiff[12]

Mathematica Complement[13]

MATLAB setdiff[14]

OCaml Set.S.diff[15]

Octave setdiff[16]

Pascal SetDifference := a - b;

Perl 5 #for perl version >= 5.10

@a = grep {not $_ ~~ @b} @a;

Perl 6 $A $B

$A (-) $B # texas version

PHP array_diff($a, $b);[17]

Prolog a(X),\+ b(X).

Python diff = a.difference(b)[18]

diff = a - b[18]

R setdiff[19]

Racket (set-subtract a b)[20]

https://en.wikipedia.org/wiki/.NET_Frameworkhttps://en.wikipedia.org/wiki/C++https://en.wikipedia.org/wiki/Clojurehttps://en.wikipedia.org/wiki/Common_Lisphttps://en.wikipedia.org/wiki/F#_(programming_language)https://en.wikipedia.org/wiki/Falcon_(programming_language)https://en.wikipedia.org/wiki/Haskell_(programming_language)https://en.wikipedia.org/wiki/Java_(programming_language)https://en.wikipedia.org/wiki/Julia_(programming_language)https://en.wikipedia.org/wiki/Mathematicahttps://en.wikipedia.org/wiki/MATLABhttps://en.wikipedia.org/wiki/OCamlhttps://en.wikipedia.org/wiki/GNU_Octavehttps://en.wikipedia.org/wiki/Pascal_(programming_language)https://en.wikipedia.org/wiki/Perl_5https://en.wikipedia.org/wiki/Perl_6https://en.wikipedia.org/wiki/PHPhttps://en.wikipedia.org/wiki/Prologhttps://en.wikipedia.org/wiki/Python_(programming_language)https://en.wikipedia.org/wiki/R_(programming_language)https://en.wikipedia.org/wiki/Racket_(programming_language)

10 CHAPTER 2. COMPLEMENT (SET THEORY)

Ruby diff = a - b[21]

Scala a.diff(b)[22]

or

a -- b[22]

Smalltalk (Pharo) a difference: b

SQL SELECT * FROM A

EXCEPT SELECT * FROM B

Unix shell comm 23 a b[23]

grep -vf b a # less efficient, but works with small unsorted sets

2.5 See also Algebra of sets

Naive set theory

Symmetric difference

2.6 References[1] Halmos (1960) p.17

[2] Devlin (1979) p.6

[3] Bourbaki p. E II.6

[4] The Comprehensive LaTeX Symbol List

[5] clojure.set API reference

[6] Common Lisp HyperSpec, Function set-difference, nset-difference. Accessed on September 8, 2009.

[7] Set.difference Method (F#). Accessed on July 12, 2015.

[9] Array subtraction, data structures. Accessed on July 28, 2014.

[10] Data.Set (Haskell)

[11] Set (Java 2 Platform SE 5.0). JavaTM 2 Platform Standard Edition 5.0 API Specification, updated in 2004. Accessed onFebruary 13, 2008.

[12] . The Standard Library--Julia Language documentation. Accessed on September 24, 2014

[13] Complement. Mathematica Documentation Center for version 6.0, updated in 2008. Accessed on March 7, 2008.

[14] Setdiff. MATLAB Function Reference for version 7.6, updated in 2008. Accessed on May 19, 2008.

[15] Set.S (OCaml).

[16] . GNU Octave Reference Manual

[17] PHP: array_diff, PHP Manual

https://en.wikipedia.org/wiki/Ruby_(programming_language)https://en.wikipedia.org/wiki/Scala_(programming_language)https://en.wikipedia.org/wiki/Smalltalk_(Pharo)https://en.wikipedia.org/wiki/SQLhttps://en.wikipedia.org/wiki/Unix_shellhttps://en.wikipedia.org/wiki/Algebra_of_setshttps://en.wikipedia.org/wiki/Naive_set_theoryhttps://en.wikipedia.org/wiki/Symmetric_differencehttp://www.lispworks.com/documentation/HyperSpec/Body/f_set_di.htmhttps://msdn.microsoft.com/en-us/library/ee340332.aspx,https://msdn.microsoft.com/en-us/library/ee353414.aspx,http://falconpl.org/index.ftd?page_id=sitewiki&prj_id=_falcon_site&sid=wiki&pwid=Survival%2520Guide&wid=Survival%253ABasic+Structures#Arrays,http://haskell.org/ghc/docs/latest/html/libraries/containers/Data-Set.htmlhttp://java.sun.com/j2se/1.5.0/docs/api/java/util/Set.htmlhttp://reference.wolfram.com/mathematica/ref/Complement.htmlhttp://www.mathworks.com/access/helpdesk/help/techdoc/ref/setdiff.htmlhttp://caml.inria.fr/pub/docs/manual-ocaml/libref/Set.S.htmlhttp://php.net/manual/en/function.array-diff.php

2.7. EXTERNAL LINKS 11

[18] . Python v2.7.3 documentation. Accessed on January 17, 2013.

[19] R Reference manual p. 410.

[20] . The Racket Reference. Accessed on May 19, 2015.

[21] Class: Array Ruby Documentation

[22] scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.

[23] comm(1), Unix Seventh Edition Manual, 1979.

Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van NostrandCompany. Zbl 0087.04403.

Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN0-387-90441-7. Zbl 0407.04003.

Bourbaki, N. (1970). Thorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.

2.7 External links Weisstein, Eric W., Complement, MathWorld.

Weisstein, Eric W., Complement Set, MathWorld.

http://cran.r-project.org/doc/manuals/fullrefman.pdfhttp://www.ruby-doc.org/core/classes/Array.htmlhttp://www.scala-lang.org/api/current/index.html#scala.collection.Set,http://plan9.bell-labs.com/7thEdMan/https://en.wikipedia.org/wiki/Paul_Halmoshttps://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0087.04403https://en.wikipedia.org/wiki/Keith_Devlinhttps://en.wikipedia.org/wiki/Springer-Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-387-90441-7https://en.wikipedia.org/wiki/Zentralblatt_MATHhttps://zbmath.org/?format=complete&q=an:0407.04003https://en.wikipedia.org/wiki/Nicolas_Bourbakihttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3-540-34034-8https://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/Complement.htmlhttps://en.wikipedia.org/wiki/MathWorldhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttp://mathworld.wolfram.com/ComplementSet.htmlhttps://en.wikipedia.org/wiki/MathWorld

Chapter 3

List of types of functions

Functions can be identified according to the properties they have. These properties describe the functions behaviourunder certain conditions. A parabola is a specific type of function.

3.1 Relative to set theory

These properties concern the domain, the codomain and the range of functions.

Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.In other words, every element of the functions codomain is the image of at most one elementof its domain.

Surjective function: has a preimage for every element of the codomain, i.e. the codomain equals the range.Also called a surjection or onto function.

Bijective function: is both an injection and a surjection, and thus invertible.

Identity function: maps any given element to itself.

Constant function: has a fixed value regardless of arguments.

Empty function: whose domain equals the empty set.

3.2 Relative to an operator (c.q. a group or other structure)

These properties concern how the function is affected by arithmetic operations on its operand.The following are special examples of a homomorphism on a binary operation:

Additive function: preserves the addition operation: f(x + y) = f(x) + f(y).

Multiplicative function: preserves the multiplication operation: f(xy) = f(x)f(y).

Relative to negation:

Even function: is symmetric with respect to the Y-axis. Formally, for each x: f(x) = f(x).

Odd function: is symmetric with respect to the origin. Formally, for each x: f(x) = f(x).

Relative to a binary operation and an order:

Subadditive function: for which the value of f(x+y) is less than or equal to f(x) + f(y).

Superadditive function: for which the value of f(x+y) is greater than or equal to f(x) + f(y).

12

https://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Domain_(mathematics)https://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Range_(mathematics)https://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Surjective_functionhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Onto_functionhttps://en.wikipedia.org/wiki/Bijective_functionhttps://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Surjectionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Identity_functionhttps://en.wikipedia.org/wiki/Constant_functionhttps://en.wikipedia.org/wiki/Empty_functionhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Group_theoryhttps://en.wikipedia.org/wiki/Mathematical_structurehttps://en.wikipedia.org/wiki/Arithmetichttps://en.wikipedia.org/wiki/Homomorphismhttps://en.wikipedia.org/wiki/Binary_operationhttps://en.wikipedia.org/wiki/Additive_maphttps://en.wikipedia.org/wiki/Multiplicative_functionhttps://en.wikipedia.org/wiki/Negationhttps://en.wikipedia.org/wiki/Even_functionhttps://en.wikipedia.org/wiki/Odd_functionhttps://en.wikipedia.org/wiki/Origin_(mathematics)https://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Subadditive_functionhttps://en.wikipedia.org/wiki/Superadditive_function

3.3. RELATIVE TO A TOPOLOGY 13

3.3 Relative to a topology Continuous function: in which preimages of open sets are open.

Nowhere continuous function: is not continuous at any point of its domain (e.g. Dirichlet function).

Homeomorphism: is an injective function that is also continuous, whose inverse is continuous.

3.4 Relative to an ordering Monotonic function: does not reverse ordering of any pair.

Strict Monotonic function: preserves the given order.

3.5 Relative to the real/complex numbers Analytic function: Can be defined locally by a convergent power series.

Arithmetic function: A function from the positive integers into the complex numbers.

Differentiable function: Has a derivative.

Smooth function: Has derivatives of all orders.

Holomorphic function: Complex valued function of a complex variable which is differentiable at every pointin its domain.

Meromorphic function: Complex valued function that is holomorphic everywhere, apart from at isolated pointswhere there are poles.

Entire function: A holomorphic function whose domain is the entire complex plane.

3.6 Ways of defining functions/Relation to Type Theory Composite function: is formed by the composition of two functions f and g, by mapping x to f(g(x)).

Inverse function: is declared by doing the reverse of a given function (e.g. arcsine is the inverse of sine).

Piecewise function: is defined by different expressions at different intervals.

In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating whatit should map to. For this purpose, the 7 symbol or Church's is often used. Also, sometimes mathematiciansnotate a functions domain and codomain by writing e.g. f : A B . These notions extend directly to lambdacalculus and type theory, respectively.

3.7 Relation to Category Theory

Category Theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms.A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a setof morphisms. A partial (equiv. dependently typed) binary operation called composition is provided on morphisms,every object has one special morphism from it to itself called the identity on that object, and composition and identitiesare required to obey certain relations.In a so-called concrete category, the objects are associated with mathematical structures like sets, magmas, groups,rings, topological spaces, vector spaces, metric spaces, partial orders, differentiable manifolds, uniform spaces, etc.,and morphisms between two objects are associated with structure-preserving functions between them. In the ex-amples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms,

https://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Open_sethttps://en.wikipedia.org/wiki/Nowhere_continuoushttps://en.wikipedia.org/wiki/Dirichlet_functionhttps://en.wikipedia.org/wiki/Homeomorphismhttps://en.wikipedia.org/wiki/Injective_functionhttps://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Analytic_functionhttps://en.wikipedia.org/wiki/Convergent_serieshttps://en.wikipedia.org/wiki/Power_serieshttps://en.wikipedia.org/wiki/Arithmetic_functionhttps://en.wikipedia.org/wiki/Integershttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Differentiable_functionhttps://en.wikipedia.org/wiki/Derivativehttps://en.wikipedia.org/wiki/Smooth_functionhttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Meromorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Pole_(complex_analysis)https://en.wikipedia.org/wiki/Entire_functionhttps://en.wikipedia.org/wiki/Holomorphic_functionhttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Composite_functionhttps://en.wikipedia.org/wiki/Inverse_functionhttps://en.wikipedia.org/wiki/Arcsinehttps://en.wikipedia.org/wiki/Sinehttps://en.wikipedia.org/wiki/Piecewise_functionhttps://en.wikipedia.org/wiki/Alonzo_Churchhttps://en.wikipedia.org/wiki/Domain_of_a_functionhttps://en.wikipedia.org/wiki/Codomainhttps://en.wikipedia.org/wiki/Lambda_calculushttps://en.wikipedia.org/wiki/Lambda_calculushttps://en.wikipedia.org/wiki/Type_theoryhttps://en.wikipedia.org/wiki/Category_Theoryhttps://en.wikipedia.org/wiki/Morphismshttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Dependently_typedhttps://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Identity_(mathematics)https://en.wikipedia.org/wiki/Concrete_categoryhttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Magmashttps://en.wikipedia.org/wiki/Group_(mathematics)https://en.wikipedia.org/wiki/Ring_(mathematics)https://en.wikipedia.org/wiki/Topological_spaceshttps://en.wikipedia.org/wiki/Vector_spaceshttps://en.wikipedia.org/wiki/Metric_spaceshttps://en.wikipedia.org/wiki/Order_theoryhttps://en.wikipedia.org/wiki/Differentiable_manifoldshttps://en.wikipedia.org/wiki/Uniform_spaceshttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Homomorphismshttps://en.wikipedia.org/wiki/Group_homomorphisms

14 CHAPTER 3. LIST OF TYPES OF FUNCTIONS

continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable func-tions, and uniformly continuous functions, respectively.As an algebraic theory, one of the advantages of category theory is to enable one to prove many general resultswith a minimum of assumptions. Many common notions from mathematics (e.g. surjective, injective, free ob-ject, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism,epimorphism).Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf.topos).Allegory theory[1] provides a generalization comparable to category theory for relations instead of functions.

3.8 References[1] Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. ISBN 978-0-

444-70368-2.

https://en.wikipedia.org/wiki/Continuous_functionshttps://en.wikipedia.org/wiki/Linear_transformationshttps://en.wikipedia.org/wiki/Matrix_(mathematics)https://en.wikipedia.org/wiki/Metric_maphttps://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Differentiablehttps://en.wikipedia.org/wiki/Uniformly_continuoushttps://en.wikipedia.org/wiki/Surjectivehttps://en.wikipedia.org/wiki/Injectivehttps://en.wikipedia.org/wiki/Free_objecthttps://en.wikipedia.org/wiki/Free_objecthttps://en.wikipedia.org/wiki/Basis_(linear_algebra)https://en.wikipedia.org/wiki/Group_representationhttps://en.wikipedia.org/wiki/Isomorphismhttps://en.wikipedia.org/wiki/Monomorphismhttps://en.wikipedia.org/wiki/Epimorphismhttps://en.wikipedia.org/wiki/Set_theoryhttps://en.wikipedia.org/wiki/Type_theoryhttps://en.wikipedia.org/wiki/Toposhttps://en.wikipedia.org/wiki/Allegory_(category_theory)https://en.wikipedia.org/wiki/Relation_(mathematics)https://en.wikipedia.org/wiki/Special:BookSources/9780444703682https://en.wikipedia.org/wiki/Special:BookSources/9780444703682

Chapter 4

Measurable function

A function is Lebesgue measurable if and only if the preimage of each of the sets [a,] is a Lebesgue measurable set.

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions betweenmeasurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function betweenmeasurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to thesituation of continuous functions between topological spaces.In probability theory, the sigma algebra often represents the set of available information, and a function (in thiscontext a random variable) is measurable if and only if it represents an outcome that is knowable based on theavailable information. In contrast, functions that are not Lebesgue measurable are generally considered pathological,at least in the field of analysis.

4.1 Formal definition

Let (X, ) and (Y, ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras and . A function f: X Y is said to be measurable if the preimage of E under f is in for every E ; i.e.

f1(E) := {x X| f(x) E} , E T.

The notion of measurability depends on the sigma algebras and . To emphasize this dependency, if f: X Y isa measurable function, we will write

f : (X,) (Y, T )

15

https://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Lebesgue_measurable_sethttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Measure_theoryhttps://en.wikipedia.org/wiki/Morphismhttps://en.wikipedia.org/wiki/Measurable_spacehttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Preimagehttps://en.wikipedia.org/wiki/Measurable_sethttps://en.wikipedia.org/wiki/Measurablehttps://en.wikipedia.org/wiki/Continuity_(topology)https://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Pathological_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysis

16 CHAPTER 4. MEASURABLE FUNCTION

4.2 Caveat

This definition can be deceptively simple, however, as special care must be taken regarding the -algebras involved.In particular, when a function f: RR is said to be Lebesgue measurable what is actually meant is that f : (R,L) (R,B) is a measurable functionthat is, the domain and range represent different -algebras on the same underlyingset (here L is the sigma algebra of Lebesgue measurable sets, and B is the Borel algebra on R). As a result, thecomposition of Lebesgue-measurable functions need not be Lebesgue-measurable.By convention a topological space is assumed to be equipped with the Borel algebra generated by its open subsetsunless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to referonly to real-valued measurable functions with respect to the Borel algebra.[1] If the values of the function lie in aninfinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weakmeasurability and Bochner measurability.

4.3 Special measurable functions If (X, ) and (Y, ) are Borel spaces, a measurable function f: (X, ) (Y, ) is also called a Borel function.Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurablefunction is nearly a continuous function; see Luzins theorem. If a Borel function happens to be a section ofsome map Y X , it is called a Borel section.

ALebesgue measurable function is a measurable function f : (R,L) (C,BC) , whereL is the sigma algebraof Lebesgue measurable sets, and BC is the Borel algebra on the complex numbers C. Lebesgue measurablefunctions are of interest in mathematical analysis because they can be integrated. In the case f : X R, f is Lebesgue measurable iff {f > } = {x X : f(x) > } is measurable for all real . This isalso equivalent to any of {f }, {f < }, {f } being measurable for all . Continuous functions,monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functionsof bounded variation are all Lebesgue measurable.[2] A function f : X C is measurable iff the real andimaginary parts are measurable.

Random variables are by definition measurable functions defined on sample spaces.

4.4 Properties of measurable functions The sum and product of two complex-valued measurable functions are measurable.[3] So is the quotient, solong as there is no division by zero.[1]

The composition of measurable functions is measurable; i.e., if f: (X, 1) (Y, 2) and g: (Y, 2) (Z,3) are measurable functions, then so is g(f()): (X, 1) (Z, 3).[1] But see the caveat regarding Lebesgue-measurable functions in the introduction.

The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) ofreal-valued measurable functions are all measurable as well.[1][4]

The pointwise limit of a sequence of measurable functions is measurable (if the codomain in endowed withthe Borel algebra); note that the corresponding statement for continuous functions requires stronger conditionsthan pointwise convergence, such as uniform convergence. (This is correct when the counter domain of theelements of the sequence is a metric space. It is false in general; see pages 125 and 126 of.[5])

4.5 Non-measurable functions

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions.

https://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Sigma_algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Real_numbershttps://en.wikipedia.org/wiki/Complex_numbershttps://en.wikipedia.org/wiki/Borel_sethttps://en.wikipedia.org/wiki/Infinite-dimensional_vector_spacehttps://en.wikipedia.org/wiki/Weak_measurabilityhttps://en.wikipedia.org/wiki/Weak_measurabilityhttps://en.wikipedia.org/wiki/Bochner_measurabilityhttps://en.wikipedia.org/wiki/Borel_spacehttps://en.wikipedia.org/wiki/Continuous_function_(topology)https://en.wikipedia.org/wiki/Luzin%2527s_theoremhttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Sigma_algebrahttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Lebesgue_integrationhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Sample_spacehttps://en.wikipedia.org/wiki/Supremumhttps://en.wikipedia.org/wiki/Infimumhttps://en.wikipedia.org/wiki/Limit_superiorhttps://en.wikipedia.org/wiki/Limit_inferiorhttps://en.wikipedia.org/wiki/Pointwise

4.6. SEE ALSO 17

So long as there are non-measurable sets in a measure space, there are non-measurable functions from thatspace. If (X, ) is some measurable space and A X is a non-measurable set, i.e. if A , then the indicatorfunction 1A: (X, ) R is non-measurable (where R is equipped with the Borel algebra as usual), since thepreimage of the measurable set {1} is the non-measurable set A. Here 1A is given by

1A(x) ={1 if x A0 otherwise

Any non-constant function can be made non-measurable by equipping the domain and range with appropriate-algebras. If f: X R is an arbitrary non-constant, real-valued function, then f is non-measurable if X isequipped with the indiscrete algebra = {, X}, since the preimage of any point in the range is some proper,nonempty subset of X, and therefore does not lie in .

4.6 See also Vector spaces of measurable functions: the Lp spaces

Measure-preserving dynamical system

4.7 Notes[1] Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.

[2] Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.

[3] Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.

[4] Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.

[5] Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.

4.8 External links Measurable function at Encyclopedia of Mathematics

Borel function at Encyclopedia of Mathematics

https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Borel_algebrahttps://en.wikipedia.org/wiki/Lp_spacehttps://en.wikipedia.org/wiki/Measure-preserving_dynamical_systemhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-7637-1497-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-49756-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-471-31716-0https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-02-404151-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-521-00754-2http://www.encyclopediaofmath.org/index.php/Measurable_functionhttp://www.encyclopediaofmath.org/http://www.encyclopediaofmath.org/index.php/Borel_functionhttp://www.encyclopediaofmath.org/

Chapter 5

Measure (mathematics)

For the coalgebra concept, see measuring coalgebra.In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of

that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length,area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assignsthe conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclideanspace Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everydaysense of the word, specifically, 1.Technically, a measure is a function that assigns a non-negative real number or + to (certain) subsets of a set X (seeDefinition below). It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset thatcan be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of thesmaller subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfyingthe other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolvedby defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are requiredto form a -algebra. This means that countable unions, countable intersections and complements of measurablesubsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be definedconsistently, are necessarily complicated in the sense of being badly mixed up with their complement.[1] Indeed, theirexistence is a non-trivial consequence of the axiom of choice.Measure theory was developed in successive stages during the late 19th and early 20th centuries by mile Borel,Henri Lebesgue, Johann Radon and Maurice Frchet, among others. The main applications of measures are in thefoundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodictheory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsetsof Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more generaland has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assignto the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

5.1 Definition

Let X be a set and a -algebra over X. A function from to the extended real number line is called a measureif it satisfies the following properties:

Non-negativity: For all E in : (E) 0.

Null empty set: () = 0.

Countable additivity (or -additivity): For all countable collections {Ei}iN of pairwise disjoint sets in :

(k=1

Ek) =k=1

(Ek)

18

https://en.wikipedia.org/wiki/Measuring_coalgebrahttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Set_(mathematics)https://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Lengthhttps://en.wikipedia.org/wiki/Areahttps://en.wikipedia.org/wiki/Volumehttps://en.wikipedia.org/wiki/Euclidean_geometryhttps://en.wikipedia.org/wiki/Dimension_(mathematics_and_physics)https://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Real_linehttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Extended_real_number_linehttps://en.wikipedia.org/wiki/Measure_(mathematics)#Definitionhttps://en.wikipedia.org/wiki/Empty_sethttps://en.wikipedia.org/wiki/Countablyhttps://en.wikipedia.org/wiki/Counting_measurehttps://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Complement_(set_theory)https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/%C3%89mile_Borelhttps://en.wikipedia.org/wiki/Henri_Lebesguehttps://en.wikipedia.org/wiki/Johann_Radonhttps://en.wikipedia.org/wiki/Maurice_Fr%C3%A9chethttps://en.wikipedia.org/wiki/Lebesgue_integralhttps://en.wikipedia.org/wiki/Andrey_Kolmogorovhttps://en.wikipedia.org/wiki/Axiomatisationhttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Riemann_integralhttps://en.wikipedia.org/wiki/Ergodic_theoryhttps://en.wikipedia.org/wiki/Dynamical_systemhttps://en.wikipedia.org/wiki/Sigma-algebrahttps://en.wikipedia.org/wiki/Extended_real_number_linehttps://en.wikipedia.org/wiki/Sigma_additivityhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Disjoint_sets

5.2. EXAMPLES 19

One may require that at least one set E has finite measure. Then the empty set automatically has measure zero becauseof countable additivity, because (E) = (E ) = (E) + () , so () = (E) (E) = 0 .If only the second and third conditions of the definition of measure above are met, and takes on at most one of thevalues , then is called a signed measure.The pair (X, ) is called ameasurable space, the members of are calledmeasurable sets. If (X,X) and (Y,Y )are two measurable spaces, then a function f : X Y is calledmeasurable if for every Y-measurable setB Y ,the inverse image is X-measurable i.e.: f (1)(B) X . The composition of measurable functions is measurable,making the measurable spaces and measurable functions a category, with the measurable spaces as objects and theset of measurable functions as arrows.A triple (X, , ) is called a measure space. A probability measure is a measure with total measure one i.e. (X)= 1. A probability space is a measure space with a probability measure.For measure spaces that are also topological spaces various compatibility conditions can be placed for the measureand the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radonmeasures. Radon measures have an alternative definition in terms of linear functionals on the locally convex spaceof continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of othersources. For more details, see the article on Radon measures.

5.2 Examples

Some important measures are listed here.

The counting measure is defined by (S) = number of elements in S.

The Lebesgue measure onR is a complete translation-invariant measure on a -algebra containing the intervalsin R such that ([0, 1]) = 1; and every other measure with these properties extends Lebesgue measure.

Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeezemapping.

The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (andalso of counting measure and circular angle measure) and has similar uniqueness properties.

The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, inparticular, fractal sets.

Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takesall its values in the unit interval [0, 1]). Such a measure is called a probability measure. See probability axioms.

The Dirac measure a (cf. Dirac delta function) is given by a(S) = S(a), where S is the characteristicfunction of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Eulermeasure, Gaussian measure, Baire measure, Radon measure, Young measure, and strong measure zero.In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negativeextensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signedmeasures,see generalizations below.Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statisticaland Hamiltonian mechanics.Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.

5.3 Properties

Several further properties can be derived from the definition of a countably additive measure.

https://en.wikipedia.org/wiki/Signed_measurehttps://en.wikipedia.org/wiki/Image_(mathematics)#Inverse_imagehttps://en.wikipedia.org/wiki/Function_compositionhttps://en.wikipedia.org/wiki/Category_(mathematics)https://en.wikipedia.org/wiki/Tuplehttps://en.wikipedia.org/wiki/Probability_measurehttps://en.wikipedia.org/wiki/Probability_spacehttps://en.wikipedia.org/wiki/Topological_spacehttps://en.wikipedia.org/wiki/Analysis_(mathematics)https://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Radon_measurehttps://en.wikipedia.org/wiki/Radon_measurehttps://en.wikipedia.org/wiki/Locally_convex_spacehttps://en.wikipedia.org/wiki/Continuous_functionhttps://en.wikipedia.org/wiki/Support_(mathematics)#Compact_supporthttps://en.wikipedia.org/wiki/Nicolas_Bourbakihttps://en.wikipedia.org/wiki/Radon_measurehttps://en.wikipedia.org/wiki/Counting_measurehttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Translational_invariancehttps://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Anglehttps://en.wikipedia.org/wiki/Rotationhttps://en.wikipedia.org/wiki/Hyperbolic_anglehttps://en.wikipedia.org/wiki/Squeeze_mappinghttps://en.wikipedia.org/wiki/Squeeze_mappinghttps://en.wikipedia.org/wiki/Haar_measurehttps://en.wikipedia.org/wiki/Locally_compact_spacehttps://en.wikipedia.org/wiki/Topological_grouphttps://en.wikipedia.org/wiki/Hausdorff_measurehttps://en.wikipedia.org/wiki/Probability_spacehttps://en.wikipedia.org/wiki/Unit_intervalhttps://en.wikipedia.org/wiki/Probability_axiomshttps://en.wikipedia.org/wiki/Dirac_measurehttps://en.wikipedia.org/wiki/Dirac_delta_functionhttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Indicator_functionhttps://en.wikipedia.org/wiki/Borel_measurehttps://en.wikipedia.org/wiki/Jordan_measurehttps://en.wikipedia.org/wiki/Ergodic_measurehttps://en.wikipedia.org/wiki/Euler_characteristichttps://en.wikipedia.org/wiki/Euler_characteristichttps://en.wikipedia.org/wiki/Gaussian_measurehttps://en.wikipedia.org/wiki/Baire_measurehttps://en.wikipedia.org/wiki/Radon_measurehttps://en.wikipedia.org/wiki/Young_measurehttps://en.wikipedia.org/wiki/Strong_measure_zerohttps://en.wikipedia.org/wiki/Masshttps://en.wikipedia.org/wiki/Gravity_potentialhttps://en.wikipedia.org/wiki/Extensive_propertyhttps://en.wikipedia.org/wiki/Conserved_quantityhttps://en.wikipedia.org/wiki/Conservation_law_(physics)https://en.wikipedia.org/wiki/Liouville%2527s_theorem_(Hamiltonian)#Symplectic_geometryhttps://en.wikipedia.org/wiki/Gibbs_measurehttps://en.wikipedia.org/wiki/Canonical_ensemble

20 CHAPTER 5. MEASURE (MATHEMATICS)

5.3.1 Monotonicity

A measure is monotonic: If E1 and E2 are measurable sets with E1 E2 then

(E1) (E2).

5.3.2 Measures of infinite unions of measurable sets

A measure is countably subadditive: For any countable sequence E1, E2, E3, ... of sets En in (not necessarilydisjoint):

( i=1

Ei

)

i=1

(Ei).

A measure is continuous from below: If E1, E2, E3, ... are measurable sets and En is a subset of En for all n,then the union of the sets En is measurable, and

( i=1

Ei

)= lim

i(Ei).

5.3.3 Measures of infinite intersections of measurable sets

A measure is continuous from above: If E1, E2, E3, ..., are measurable sets and for all n, En En, then theintersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

( i=1

Ei

)= lim

i(Ei).

This property is false without the assumption that at least one of the En has finite measure. For instance, for each n N, let En = [n, ) R, which all have infinite Lebesgue measure, but the intersection is empty.

5.4 Sigma-finite measures

Main article: Sigma-finite measure

A measure space (X, , ) is called finite if (X) is a finite real number (rather than ). Nonzero finite measures areanalogous to probability measures in the sense that any finite measure is proportional to the probability measure

1(X) . A measure is called -finite if X can be decomposed into a countable union of measurable sets of finitemeasure. Analogously, a set in a measure space is said to have a -finite measure if it is a countable union of sets withfinite measure.For example, the real numbers with the standard Lebesgue measure are -finite but not finite. Consider the closedintervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union isthe entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finiteset of reals the number of points in the set. This measure space is not -finite, because every set with finite measurecontains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The-finite measure spaces have some very convenient properties; -finiteness can be compared in this respect to theLindelf property of topological spaces. They can be also thought of as a vague generalization of the idea that ameasure space may have 'uncountable measure'.

https://en.wikipedia.org/wiki/Monotonic_functionhttps://en.wikipedia.org/wiki/Subadditivityhttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Sequence_(mathematics)https://en.wikipedia.org/wiki/Union_(set_theory)https://en.wikipedia.org/wiki/Intersection_(set_theory)https://en.wikipedia.org/wiki/Sigma-finite_measurehttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Closed_intervalhttps://en.wikipedia.org/wiki/Closed_intervalhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Counting_measurehttps://en.wikipedia.org/wiki/Lindel%C3%B6f_space

5.5. COMPLETENESS 21

5.5 Completeness

Main article: Complete measure

A measurable set X is called a null set if (X) = 0. A subset of a null set is called a negligible set. A negligible setneed not be measurable, but every measurable negligible set is automatically a null set. A measure is called completeif every negligible set is measurable.A measure can be extended to a complete one by considering the -algebra of subsets Y which differ by a negligibleset from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. Onedefines (Y) to equal (X).

5.6 Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any setI and any set of nonnegative ri, i I define:

iI

ri = sup{

iJri : |J | < 0, J I

}.

That is, we define the sum of the ri to be the supremum of all the sums of finitely many of them.A measure on is -additive if for any < and any family X , < the following hold:

X

(

X

)=

(X) .

Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.

5.7 Non-measurable sets

Main article: Non-measurable set

If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examplesof such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the BanachTarski paradox.

5.8 Generalizations

For certain purposes, it is useful to have a measure whose values are not restricted to the non-negative reals orinfinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signedmeasure, while such a function with values in the complex numbers is called a complex measure. Measures that takevalues in Banach spaces have been studied extensively.[2] A measure that takes values in the set of self-adjoint pro-jections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectraltheorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations,the term positive measure is used. Positive measures are closed under conical combination but not general linearcombination, while signed measures are the linear closure of positive measures.Another generalization is the finitely additive measure, which are sometimes called contents. This is the same asa measure except that instead of requiring countable additivity we require only finite additivity. Historically, this

https://en.wikipedia.org/wiki/Complete_measurehttps://en.wikipedia.org/wiki/Null_sethttps://en.wikipedia.org/wiki/Symmetric_differencehttps://en.wikipedia.org/wiki/Ideal_(set_theory)https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Euclidean_spacehttps://en.wikipedia.org/wiki/Lebesgue_measurablehttps://en.wikipedia.org/wiki/Vitali_sethttps://en.wikipedia.org/wiki/Hausdorff_paradoxhttps://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradoxhttps://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradoxhttps://en.wikipedia.org/wiki/Set_functionhttps://en.wikipedia.org/wiki/Signed_measurehttps://en.wikipedia.org/wiki/Signed_measurehttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Complex_measurehttps://en.wikipedia.org/wiki/Banach_spaceshttps://en.wikipedia.org/wiki/Hilbert_spacehttps://en.wikipedia.org/wiki/Projection-valued_measurehttps://en.wikipedia.org/wiki/Functional_analysishttps://en.wikipedia.org/wiki/Spectral_theoremhttps://en.wikipedia.org/wiki/Spectral_theoremhttps://en.wikipedia.org/wiki/Conical_combinationhttps://en.wikipedia.org/wiki/Linear_combinationhttps://en.wikipedia.org/wiki/Linear_combinationhttps://en.wikipedia.org/wiki/Content_(measure_theory)

22 CHAPTER 5. MEASURE (MATHEMATICS)

definition was used first. It turns out that in general, finitely additive measures are connected with notions such asBanach limits, the dual of L and the Stoneech compactification. All these are linked in one way or another to theaxiom of choice.A charge is a generalization in both directions: it is a finitely additive, signed measure.

5.9 See also

Abelian von Neumann algebra

Almost everywhere

Carathodorys extension theorem

Fubinis theorem

Fatous lemma

Fuzzy measure theory

Geometric measure theory

Hausdorff measure

Inner measure

Lebesgue integration

Lebesgue measure

Lorentz space

Lifting theory

Measurable function

Outer measure

Product measure

Pushforward measure

Vector measure

Volume form

Measurable cardinal

5.10 References

[1] Halmos, Paul (1950), Measure theory, Van Nostrand and Co.

[2] Rao, M. M. (2012), Random and vector measures, Series on Multivariate Analysis 9, World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, ISBN 978-981-4350-81-5, MR 2840012.

https://en.wikipedia.org/wiki/Banach_limithttps://en.wikipedia.org/wiki/Lp_spacehttps://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactificationhttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Signed_measurehttps://en.wikipedia.org/wiki/Abelian_von_Neumann_algebrahttps://en.wikipedia.org/wiki/Almost_everywherehttps://en.wikipedia.org/wiki/Carath%C3%A9odory%2527s_extension_theoremhttps://en.wikipedia.org/wiki/Fubini%2527s_theoremhttps://en.wikipedia.org/wiki/Fatou%2527s_lemmahttps://en.wikipedia.org/wiki/Fuzzy_measure_theoryhttps://en.wikipedia.org/wiki/Geometric_measure_theoryhttps://en.wikipedia.org/wiki/Hausdorff_measurehttps://en.wikipedia.org/wiki/Inner_measurehttps://en.wikipedia.org/wiki/Lebesgue_integrationhttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Lorentz_spacehttps://en.wikipedia.org/wiki/Lifting_theoryhttps://en.wikipedia.org/wiki/Measurable_functionhttps://en.wikipedia.org/wiki/Outer_measurehttps://en.wikipedia.org/wiki/Product_measurehttps://en.wikipedia.org/wiki/Pushforward_measurehttps://en.wikipedia.org/wiki/Vector_measurehttps://en.wikipedia.org/wiki/Volume_formhttps://en.wikipedia.org/wiki/Measurable_cardinalhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-981-4350-81-5https://en.wikipedia.org/wiki/Mathematical_Reviewshttps://www.ams.org/mathscinet-getitem?mr=2840012

5.11. BIBLIOGRAPHY 23

5.11 Bibliography Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.

Bauer, H. (2001), Measure and Integration Theory, Berlin: de Gruyter, ISBN 978-3110167191

Bear, H.S. (2001), A Primer of Lebesgue Integration, San Diego: Academic Press, ISBN 978-0120839711

Bogachev, V. I. (2006), Measure theory, Berlin: Springer, ISBN 978-3540345138

Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.

R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.

Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons,ISBN 0471317160 Second edition.

D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.

Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag,ISBN 3-540-44085-2

R. Duncan Luce and Louis Narens (1987). measurement, theory of, The New Palgrave: A Dictionary ofEconomics, v. 3, pp. 42832.

M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.

K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of Charges: A Study of Finitely Additive Measures,London: Academic Press, pp. x + 315, ISBN 0-12-095780-9

Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A.Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.

Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)

Terence Tao, 2011. An Introduction to Measure Theory. American Mathematical Society.

Nik Weaver, 2013. Measure Theory and Functional Analysis. World Scientific Publishing.

5.12 External links Hazewinkel, Michiel, ed. (2001), Measure, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Tutorial: Measure Theory for Dummies

https://en.wikipedia.org/wiki/Robert_G._Bartlehttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3110167191https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-0120839711https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-3540345138https://en.wikipedia.org/wiki/Springer_Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-41129-1https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0471317160http://www.essex.ac.uk/maths/people/fremlin/mt.htmhttps://en.wikipedia.org/wiki/Springer_Verlaghttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-44085-2https://en.wikipedia.org/wiki/R._Duncan_Lucehttps://en.wikipedia.org/wiki/New_Palgrave:_A_Dictionary_of_Economicshttps://en.wikipedia.org/wiki/New_Palgrave:_A_Dictionary_of_Economicshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-12-095780-9https://en.wikipedia.org/wiki/Special:BookSources/0486635198https://en.wikipedia.org/wiki/Daniell_integralhttps://en.wikipedia.org/wiki/Gerald_Teschlhttp://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.htmlhttps://en.wikipedia.org/wiki/Terence_Taohttps://en.wikipedia.org/wiki/Nik_Weaverhttp://www.encyclopediaofmath.org/index.php?title=p/m063240https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematicshttps://en.wikipedia.org/wiki/Springer_Science+Business_Mediahttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4https://en.wikipedia.org/wiki/Special:BookSources/978-1-55608-010-4http://www.ee.washington.edu/techsite/papers/documents/UWEETR-2006-0008.pdf

24 CHAPTER 5. MEASURE (MATHEMATICS)

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than orequal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

https://en.wikipedia.org/wiki/Monotone_functionhttps://en.wikipedia.org/wiki/Subsethttps://en.wikipedia.org/wiki/Empty_set

5.12. EXTERNAL LINKS 25

( ) ( ) ( )( )

= ++ + ...

Countable additivity of a measure : The measure of a countable disjoint union is the same as the sum of all measures of eachsubset.

Chapter 6

Vitali set

In mathematics, aVitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, foundby Vitali[1]. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitalisets, and their existence is proven on the assumption of the axiom of choice.

6.1 Measurable sets

Certain sets have a definite 'length' or 'mass. For instance, the interval [0, 1] is deemed to have length 1; moregenerally, an interval [a, b], a b, is deemed to have length ba. If we think of such intervals as metal rods withuniform density, they likewise have well-defined masses. The set [0, 1] [2, 3] is composed of two intervals of lengthone, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass or 'total length'? Asan example, we might ask what is the mass of the set of rational numbers, given that the mass of the interval [0, 1]is 1. The rationals are dense in the reals, so any non negative value may appear reasonable.However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assignsa measure of b a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because itis countable. Any set which has a well-defined Lebesgue measure is said to be measurable, but the constructionof the Lebesgue measure (for instance using Carathodorys extension theorem) does not make it obvious whethernon-measurable sets exist. The answer to that question involves the axiom of choice.

6.2 Construction and proof

A Vitali set is a subset V of the interval [0, 1] of real numbers such that, for each real number r, there is exactly onenumber v V such that vr is a rational number. Vitali sets exist because the rational numbers Q form a normalsubgroup of the real numbersR under addition, and this allows the construction of the additive quotient groupR/Q ofthese two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbersunder addition. This group R/Q consists of disjoint shifted copies of the rational numbers in the sense that eachelement of this quotient group is a set of the form Q + r for some r in R. The uncountably many elements of R/QpartitionR, and each element is dense inR. Each element ofR/Q intersects [0, 1], and the axiom of choice guaranteesthe existence of a subset of [0, 1] containing exactly one representative out of each element of R/Q. A set formedthis way is called a Vitali set.Every Vitali set V is uncountable, and vu is irrational for any u, v V, u = v .A Vitali set is non-measurable. To show this, we assume that V is measurable and we derive a contradiction. Let q1,q2, ... be an enumeration of the rational numbers in [1, 1] (recall that the rational numbers are countable). From theconstruction of V, note that the translated sets Vk = V + qk = {v + qk : v V } , k = 1, 2, ... are pairwise disjoint,and further note that [0, 1]

k Vk [1, 2] . (To see the first inclusion, consider any real number r in [0, 1] and

let v be the representative in V for the equivalence class [r]; then rv = q for some rational number q in [1, 1].)Apply the Lebesgue measure to these inclusions using sigma additivity:

26

https://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Existence_theoremhttps://en.wikipedia.org/wiki/Uncountably_manyhttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Dense_sethttps://en.wikipedia.org/wiki/Sigma_additivityhttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Carath%C3%A9odory%2527s_extension_theoremhttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Interval_(mathematics)https://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Rational_numberhttps://en.wikipedia.org/wiki/Normal_subgrouphttps://en.wikipedia.org/wiki/Normal_subgrouphttps://en.wikipedia.org/wiki/Additionhttps://en.wikipedia.org/wiki/Quotient_grouphttps://en.wikipedia.org/wiki/Cosethttps://en.wikipedia.org/wiki/Disjoint_setshttps://en.wikipedia.org/wiki/Uncountable_sethttps://en.wikipedia.org/wiki/Partition_of_a_sethttps://en.wikipedia.org/wiki/Dense_sethttps://en.wikipedia.org/wiki/Axiom_of_choicehttps://en.wikipedia.org/wiki/Countablehttps://en.wikipedia.org/wiki/Sigma_additivity

6.3. SEE ALSO 27

1 k=1

(Vk) 3.

Because the Lebesgue measure is translation invariant, (Vk) = (V ) and therefore

1 k=1

(V ) 3.

But this is impossible. Summing infinitely many copies of the constant (V) yields either zero or infinity, accordingto whether the constant is zero or positive. In neither case is the sum in [1, 3]. So V cannot have been measurableafter all, i.e., the Lebesgue measure must not define any value for (V).

6.3 See also Non-measurable set

BanachTarski paradox

6.4 References[1] Vitali, Giuseppe (1905). Sul problema della misura dei gruppi di punti di una retta. Bologna, Tip. Gamberini e Parmeg-

giani.

6.5 Bibliography Herrlich, Horst (2006). Axiom of Choice. Springer. p. 120.

Vitali, Giuseppe (1905). Sul problema della misura dei gruppi di punti di una retta. Bologna, Tip. Gamberinie Parmeggiani.

https://en.wikipedia.org/wiki/Non-measurable_sethttps://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradoxhttps://en.wikipedia.org/wiki/Giuseppe_Vitalihttps://en.wikipedia.org/wiki/Giuseppe_Vitali

28 CHAPTER 6. VITALI SET

6.6 Text and image sources, contributors, and licenses

6.6.1 Text Algebra of sets Source: https://en.wikipedia.org/wiki/Algebra_of_sets?oldid=660214875 Contributors: The Anome, AugPi, Charles

Matthews, WhisperToMe, Wile E. Heresiarch, Tobias Bergemann, Macrakis, Doshell, Discospinster, Esse~enwiki, Slipstream, Paul Au-gust, Oleg Alexandrov, Woohookitty, Isnow, Salix alba, Juan Marquez, Mathbot, Splintercellguy, Trovatore, Arthur Rubin, Gilliam,Bluebot, Javalenok, Byelf2007, Jackzhp, Kupirijo, Josephpetty100, Policron, The enemies of god, Alex10023, Jamelan, Tcamps42, An-chor Link Bot, Hans Adler, Addbot, Legobot, TaBOT-zerem, Matt Popat, Bluerasberry, Corruptcopper, Saeidpourbabak, Yahia.barie,Specs112, MegaSloth, DASHBot, Set theorist, ClueBot NG, MerlIwBot, DonMTobin, Helpful Pixie Bot, Daviddwd, Palltrast, Chris-Gualtieri, Freeze S, Jochen Burghardt, YiFeiBot, Eniacpx, Plcarmelbiron and Anonymous: 41

Complement (set theory) Source: https://en.wikipedia.org/wiki/Complement_(set_theory)?oldid=671147118 Contributors: DamianYerrick, AxelBoldt, Fnielsen, Toby Bartels, Michael Hardy, Wshun, 6birc, Julesd, AugPi, Charles Matthews, Dcoetzee, Wik, E23~enwiki,Hyacinth, Robbot, Peak, Bkell, Anthony, Tobias Bergemann, Giftlite, Lethe, Elektron, SamHocevar, TheObtuseAngleOfDoom, Running,Rich Farmbrough, Paul August, Robertbowerman, , EmilJ, Nk, Mroach, Caesura, Oleg Alexandrov, Mindmatrix, Shreevatsa, Isnow,Salix alba, JuanMarquez, Brighterorange, Tardis, BMF81, Chobot, YurikBot, RussBot, Trovatore, Nathan11g, Haginile~enwiki, Tomisti,Arthur Rubin, Mike1024, SmackBot, Melchoir, Deon Steyn, BiT, Bluebot, Georgelulu, Octahedron80, DHN-bot~enwiki, Acepectif, Pi-lotguy, Newone, CBM, Pjvpjv, Escarbot, JAnDbot, Kcho, VoABot II, Greg Ward, Gja822, JCraw, CommonsDelinker, Pharaoh of theWizards, Numbo3, Ttwo, Pasixxxx, Kyle the bot, Hqb, Philmac, AlleborgoBot, Haiviet~enwiki, BotMultichill, Jerryobject, Yoda ofBorg, Anchor Link Bot, DEMcAdams, ClueBot, Dead10ck, Ideal gas equation, Alexbot, Watchduck, Ewger, Addbot, Luckas-bot, Yobot,Nallimbot, AnomieBOT, Ciphers, Jim1138, Txebixev, Martnym, RibotBOT, Entropeter, Thehelpfulbot, FrescoBot, Paine Ellsworth,Pinethicket, TobeBot, Lotusoneny, Whisky drinker, 1qaz-pl, EmausBot, AsceticRose, ZroBot, Alpha Quadrant (alt), Nxtfari, Minoru-kun, ClueBot NG, Bgcaus, MelbourneStar, Pzrq, Catch2011, Chmarkine, Brad7777, Bakkedal, Freeze S, Deltahedron, Lambda Fairy,Manman2323, Gpendergast, Bradleyscollins and Anonymous: 78

List of types of functions Source: https://en.wikipedia.org/wiki/List_of_types_of_functions?oldid=666504554 Contributors: Zundark,Tompw, RussBot, JoannaSerah, R'n'B, Muhandes, SchreiberBike, Addbot, PV=nRT, Luckas-bot, Quondum, Bomazi, ClueBot NG,Brad7777, Kurtbusch2, Yogendra kumar Pal and Anonymous: 7

Measurable function Source: https://en.wikipedia.org/wiki/Measurable_function?oldid=627027644 Contributors: AxelBoldt, Zundark,Miguel~enwiki, Imran, Silverfish, Charles Matthews, Dysprosia, Jitse Niesen, Fibonacci, Robbot, Weialawaga~enwiki, Tosha, Giftlite,Rs2, Fropuff, CSTAR,Maneesh, Vivacissamamente, Paul August, Msh210, Oleg Alexandrov, Linas, OdedSchramm, Mandarax, MarSch,Salix alba, Mike Segal, Chobot, RussBot, GeeJo, Dan131m, Zvika, SmackBot, Gala.martin, Chungc, Richard L. Peterson, Vanisheduser v8n3489h3tkjnsdkq30u3f, Pascal.Tesson, Thijs!bot, GiM, Salgueiro~enwiki, Takwan, STBot, Jahredtobin, Jka02, AlleborgoBot,SieBot, COBot, Rinconsoleao, Dingenis, ClculIntegral, Addbot, Loewepeter, PV=nRT, Yobot, Ht686rg90, Calle, Bdmy, GrouchoBot,Undsoweiter, Sawomir Biay, Mjs1991, Walterfm, Ron asquith, EmausBot, ZroBot, Parodi, Helpful Pixie Bot, Brirush, LHSPhantomand Anonymous: 36

Measure (mathematics) Source: https://en.wikipedia.org/wiki/Measure_(mathematics)?oldid=669888044Contributors: AxelBoldt, Zun-dark, Iwnbap, Andre Engels, Toby~enwiki, Toby Bartels, Miguel~enwiki, Patrick, Michael Hardy, Gabbe, TakuyaMurata, Loisel, LorenRosen, Revolver, Charles Matthews, Dino, Dysprosia, Prumpf, Fibonacci, Robbot, Gandalf61, MathMartin, Sverdrup, Aetheling, Ru-akh, Tobias Bergemann, Pdenapo, Weialawaga~enwiki, Ancheta Wis, Tosha, Giftlite, Mousomer, BenFrantzDale, Lethe, Lupin, Math-Knight, Everyking, Mike40033, Uranographer, OverlordQ, CSTAR, Pmanderson, Vivacissamamente, PhotoBox, Keenanpepper, Dis-cospinster, Rich Farmbrough, Gadykozma, Mat cross, Harriv, Paul August, Bender235, Elwikipedista~enwiki, Gauge, Rgdboer, Jungdalglish, Gar37bic, 3mta3, Obradovic Goran, Tsirel, Msh210, Alansohn, ABCD, Caesura, Jheald, AiusEpsi, Oleg Alexandrov, Feezo,Joriki, DealPete, Linas, Isnow, BD2412, Dpv, MarSch, Salix alba, Mike Segal, R.e.b., FlaBot, Mathbot, Chobot, DVdm, Bgwhite,Roboto de Ajvol, YurikBot, RMcGuigan, Manop, Gaius Cornelius, Rat144, Crasshopper, Bota47, Rktect, Googl, Brian Tvedt, Mebden,Benandorsqueaks, GrinBot~enwiki, Zvika, Finell, SmackBot, Eskimbot, Gilliam, SchfiftyThree, MrRage, RayAYang, Nbarth, DHN-bot~enwiki, Geevee, Foxjwill, Turms, EIFY, Henning Makholm, Jna runn, Lambiam, Richard L. Peterson, The Infidel, Irvin83,16@r, Ashigabou, Levineps, Zero sharp, Markjoseph125, CRGreathouse, Jackzhp, CBM, Matthew Auger, Thomasmeeks, Myasuda,Xantharius, Omicronpersei8, Thijs!bot, Edokter, Salgueiro~enwiki, JAnDbot, MER-C, Takwan, Avaya1, Arvinder.virk, Beaumont, Ro-gierBrussee, Jay Gatsby, Albmont, GIrving, Sullivan.t.j, David Eppstein, Akulo, Cdamama, Daniele.tampieri, Policron, Juliancolton,VolkovBot, Hesam7, Saibod, Digby Tantrum, PaulTanenbaum, Geometry guy, Piyush Sriva, Ocsenave, SieBot, Stca74, Boobahmad101,BrianS36, Paolo.dL, Mimihitam, Thehotelambush, Jorgen W, Baaaaaaar, S2000magician, Melcombe, Ken123BOT, The Thing ThatShould Not Be, MABadger, Lbertolotti, Masterpiece2000, Sun Creator, 7&6=thirteen, Nicoguaro, SilvonenBot, Addbot, AndersBot,Yobot, AnomieBOT, Bdmy, Dowjgyta, Ptrf, Jsharpminor, Omnipaedista, Point-set topologist, Charvest, Semistablesystem, FrescoBot,Dave Ordinary, Sawomir Biay, Citation bot 1, Zhangkai Jason Cheng, Dark Charles, RandomDSdevel, Kiefer.Wolfowitz, Yahia.barie,Danielbojczuk, Pokus9999, Tcnuk, Bgpaulus, Le Docteur, AleHitch, Boplin, Slawekb, Empty Buffer, ClueBot NG, Michael P. Bar-nett, Frietjes, Joel B. Lewis, Finanzmaster, MerlIwBot, Helpful Pixie Bot, KLBot2, Boriaj, HilberTraum, Brad7777, Randomguess,ChrisGualtieri, Avastration, Acehole60, Stephan Kulla, Limit-theorem, Mark viking, Tentinator, Clebor42, K9re11, Suelru, KasparBot,Ganatuiyop and Anonymous: 130

Vitali set Source: https://en.wikipedia.org/wiki/Vitali_set?oldid=665953875Contributors: Michael Hardy, Loisel, Cyan, Revolver, CharlesMatthews, Dino, Fibonacci, Bkell, UtherSRG,Aetheling, JerryFriedman, Tobias Bergemann, Tosha, Giftlite, IanMaxwell, SimonLacoste-Julien, Vivacissamamente, Gadykozma, ArnoldReinhold, Crisfilax, Touriste, Army1987, BernardH, Oleg Alexandrov, Linas, Rictus,Hgkamath, R.e.b., Chobot, YurikBot, Archelon, AlexeiK, Petter Strandmark, Trovatore, RDBury, Karl Stroetmann, Lim Wei Quan,TooMuchMath, W3asal, Salgueiro~enwiki, JJ Harrison, Sullivan.t.j, Friday529, Polkaparty, PaulTanenbaum, Likebox, MaSt, Addbot,PV=nRT, Luckas-bot, Yobot, LGB, Piano non troppo, Constructive editor, Mafaraxas, Conjugado, ZroBot, WeijiBaikeBianji, MaxLongint, Makecat-bot, Zamir1234 and Anonymous: 36

6.6.2 Images File:Countable_additivity_of_a_measure.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Countable_additivity_

of_a_measure.svg License: CC BY 3.0 Contributors: Own work Original artist: Stephan Kulla (User:Stephan Kulla)

https://en.wikipedia.org/wiki/Algebra_of_sets?oldid=660214875https://en.wikipedia.org/wiki/Complement_(set_theory)?ol