ELEMENTARY MEASURE THEORY

Let S be a set. A collection B0 of subsets of S is called an algebra iff:

1. If A, A' ∈ B0, then A ∩ A' ∈ B0 and A ∪ A' ∈ B0.

2. If A ∈ B0, then S\A ∈ B0.

3. S ∈ B0.

A collection B is called a σ-algebra if it is an algebra and, in addition,

1. if {An} is a countable collection of subsets of S such that, for every n, An ∈ B, then ∪nAn ∈ B.

A measurable space is a structure 〈S,B〉, where S is a set and B is a σ-algebra of subsets of S.

Let 〈S,B〉 be a measurable space. A finitely additive measure is a function m: B→[0,∞) such that:

1. If A,A' ∈ B and A ∩ A' = Ø, then m(A ∪ A') = m(A) + m(A').

2. m(Ø) = 0.

If m(S) < ∞, m is a finite measure. If m(S) = 1 m is a normalized probability measure.

m is a measure if it is a finitely additive measure and, in addition,

1. If {An} ⊆ B is a countable collection of mutually exclusive measurable sets, then m(∪nAn) = Σnm(An) (σ-additivity).

〈S,B,m〉 is called a measure space.

Theorem (Caratheodory). If m is a σ-additive measure on an algebra B0, it can be extended, uniquely, to a measure on the σ-algebra that extends B0.