proof of Carathéodory’s lemma
A set S⊆X is μ-measurable if and only if
μ(E)≥μ(E∩S)+μ(E∩Sc) | (1) |
for every E⊆X.
As this inequality is clearly satisfied if S=∅ and is unchanged when S is replaced by Sc, then 𝒜 contains the empty set
and is closed under taking complements
of sets.
To show that 𝒜 is a σ-algebra, it only remains to show that it is closed under taking countable
unions of sets. Choose any sets A,B∈𝒜 and E⊆X. Then,
μ(E)≥μ(E∩A)+μ(E∩Ac)≥μ(E∩A)+μ(E∩Ac∩B)+μ(E∩Ac∩Bc)≥μ(E∩(A∪B))+μ(E∩Ac∩Bc) |
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∪(Ac∩B)=A∪B. So (1) is satisfied with A∪B in place of S, showing that 𝒜 is closed under taking pairwise unions and is therefore an algebra of sets on X. If A,B are disjoint sets in 𝒜 then replacing E by E∩(A∪B) and S by A in (1) gives μ(E∩(A∪B))≥μ(E∩A)+μ(E∩B). As the reverse inequality follows from subadditivity of μ, this implies that
μ(E∩(A∪B))=μ(E∩A)+μ(E∩B). |
So, the map A↦μ(E∩A) is an additive set function on 𝒜. In particular, taking E=X shows that μ is additive on 𝒜.
Now choose a sequence Ai∈𝒜, and set Bi≡⋃ij=1Aj which are in the algebra 𝒜. To prove that 𝒜 is a σ-algebra it needs to be shown that A≡⋃iAi=⋃iBi is itself in 𝒜. First, as Bi∈𝒜 and Ac⊆Bci,
μ(E)≥μ(E∩Bi)+μ(E∩Bci)≥μ(E∩Bi)+μ(E∩Ac). |
As Ci≡Bi∖Bi-1 are pairwise disjoint sets in 𝒜 satisfying ⋃ij=1Cj=Bi the additivity of C↦μ(E∩C) on 𝒜 gives
μ(E)≥i∑j=1μ(E∩Cj)+μ(E∩Ac). |
So, letting i increase to infinity, the subadditivity of μ applied to ⋃j(E∩Cj)=E∩A gives
μ(E)≥∑jμ(E∩Cj)+μ(E∩Ac)≥μ(E∩A)+μ(E∩Ac). |
This shows that A is μ-measurable and so 𝒜 is a σ-algebra.
It only remains to show that the restriction of μ to 𝒜 is a measure
, for which it needs to be shown that μ is countably additive on 𝒜. So, choose any pairwise disjoint sequence Ai∈𝒜 and set A=⋃iAi. The following inequality
i∑j=1μ(Aj)=μ(i⋃j=1Aj)≤μ(A)≤∑jμ(Aj) |
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 |