proof of Carathéodory’s lemma

A set SX is μ-measurable if and only if

μ(E)μ(ES)+μ(ESc) (1)

for every EX. As this inequalityMathworldPlanetmath is clearly satisfied if S= and is unchanged when S is replaced by Sc, then 𝒜 contains the empty setMathworldPlanetmath and is closed under taking complementsMathworldPlanetmath of sets. To show that 𝒜 is a σ-algebra, it only remains to show that it is closed under taking countableMathworldPlanetmath unions of sets. Choose any sets A,B𝒜 and EX. Then,


The first two inequalities here follow from applying (1) with A and then B in place of S, and the third uses the subadditivity of μ together with A(AcB)=AB. So (1) is satisfied with AB in place of S, showing that 𝒜 is closed under taking pairwise unions and is therefore an algebra of setsMathworldPlanetmath on X. If A,B are disjoint sets in 𝒜 then replacing E by E(AB) and S by A in (1) gives μ(E(AB))μ(EA)+μ(EB). As the reverse inequality follows from subadditivity of μ, this implies that


So, the map Aμ(EA) is an additive set function on 𝒜. In particular, taking E=X shows that μ is additive on 𝒜.

Now choose a sequence Ai𝒜, and set Bij=1iAj which are in the algebra 𝒜. To prove that 𝒜 is a σ-algebra it needs to be shown that AiAi=iBi is itself in 𝒜. First, as Bi𝒜 and AcBic,


As CiBiBi-1 are pairwise disjoint sets in 𝒜 satisfying j=1iCj=Bi the additivity of Cμ(EC) on 𝒜 gives


So, letting i increase to infinityMathworldPlanetmathPlanetmath, the subadditivity of μ applied to j(ECj)=EA gives


This shows that A is μ-measurable and so 𝒜 is a σ-algebra.

It only remains to show that the restrictionPlanetmathPlanetmathPlanetmath of μ to 𝒜 is a measureMathworldPlanetmath, for which it needs to be shown that μ is countably additive on 𝒜. So, choose any pairwise disjoint sequence Ai𝒜 and set A=iAi. The following inequality


follows from the additivity of μ on 𝒜, the requirement that μ is increasing and from the countable subadditivity of μ. Letting i increase to infinity gives μ(A)=jμ(Aj) and μ is indeed countably additive on 𝒜.

Title proof of Carathéodory’s lemma
Canonical name ProofOfCaratheodorysLemma
Date of creation 2013-03-22 18:33:25
Last modified on 2013-03-22 18:33:25
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Proof
Classification msc 28A12
Related topic CaratheodorysLemma
Related topic OuterMeasure