A σ-field F is simply a class of sets with three properties.  σ-fields are important for probability theory

because those properties ensure that Fcontains all of the sets to which the axioms of probability refer when

applied to elements of F.

Let Ω be a partition of the sure event. Let F be a class of subsets of Ω.

The Axioms of Probability

1. For all A in F, Pr(A)≥0.

2. Pr(Ω)=13. For all sequences A1,A2,… of disjoint sets in F, Pr(∪∞k=1Ak)=∑∞k=1Pr(Ak).

The Properties of a σ-Field

1. Axiom 2 refers to Ω, so F must contain Ω.

2. For every A in F, Axioms 2 and 3 together imply Pr(Ac)=1−Pr(A).1  Thus, F must

contain Ac if it contains A.

3. Axiom 3 refers to ∪∞k=1Ak, so F must contain this set if it contains A1,A2,….

More compactly, a σ-field is a class of sets that contains the sure event and is closed under complementation

and countable union.

1. This result follows from the fact that Axiom 2 says Pr(Ω)=Pr(A∪Ac)=1, and Axiom 3

says Pr(A∪Ac)=Pr(A)+Pr(Ac).

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