14.2. CARATHEODORY EXTENSION THEOREM 387

14.2 Caratheodory Extension TheoremThe Caratheodory extension theorem is a fundamental result which makes possible theconsideration of measures on infinite products among other things. The idea is that if afinite measure defined only on an algebra is trying to be a measure, then in fact it can beextended to a measure.

Definition 14.2.1 Let E be an algebra of sets of Ω and let µ0 be a finite measure on E .This means µ0 is finitely additive and if Ei,E are sets of E with the Ei disjoint and

E = ∪∞i=1Ei,

then

µ0 (E) =∞

∑i=1

µ0 (Ei)

while µ0 (Ω)< ∞.

In this definition, µ0 is trying to be a measure and acts like one whenever possible.Under these conditions, µ0 can be extended uniquely to a complete measure, µ , defined ona σ algebra of sets containing E such that µ agrees with µ0 on E . The following is themain result.

Theorem 14.2.2 Let µ0 be a measure on an algebra of sets, E , which satisfies µ0 (Ω)< ∞.Then there exists a complete measure space (Ω,S , µ) such that

µ (E) = µ0 (E)

for all E ∈ E . Also if ν is any such measure which agrees with µ0 on E , then ν = µ onσ (E ), the σ algebra generated by E .

Proof: Define an outer measure as follows.

µ (S)≡ inf

{∞

∑i=1

µ0 (Ei) : S⊆ ∪∞i=1Ei,Ei ∈ E

}Claim 1: µ is an outer measure.Proof of Claim 1: Let S⊆ ∪∞

i=1Si and let Si ⊆ ∪∞j=1Ei j, where

µ (Si)+ε

2i ≥∞

∑j=1

µ (Ei j) .

Thenµ (S)≤∑

i∑

jµ (Ei j) = ∑

i

(µ (Si)+

ε

2i

)= ∑

iµ (Si)+ ε.

Since ε is arbitrary, this shows µ is an outer measure as claimed.By the Caratheodory procedure, there exists a unique σ algebra, S , consisting of the

µ measurable sets such that(Ω,S , µ)