proof of Carathéodory’s lemma
A set is -measurable if and only if
for every . As this inequality is clearly satisfied if and is unchanged when is replaced by , then contains the empty set and is closed under taking complements of sets. To show that is a -algebra, it only remains to show that it is closed under taking countable unions of sets. Choose any sets and . Then,
The first two inequalities here follow from applying (1) with and then in place of , and the third uses the subadditivity of together with . So (1) is satisfied with in place of , showing that is closed under taking pairwise unions and is therefore an algebra of sets on . If are disjoint sets in then replacing by and by in (1) gives . As the reverse inequality follows from subadditivity of , this implies that
So, the map is an additive set function on . In particular, taking shows that is additive on .
Now choose a sequence , and set which are in the algebra . To prove that is a -algebra it needs to be shown that is itself in . First, as and ,
As are pairwise disjoint sets in satisfying the additivity of on gives
So, letting increase to infinity, the subadditivity of applied to gives
This shows that is -measurable and so is a -algebra.
It only remains to show that the restriction of to is a measure, for which it needs to be shown that is countably additive on . So, choose any pairwise disjoint sequence and set . The following inequality
|Title||proof of Carathéodory’s lemma|
|Date of creation||2013-03-22 18:33:25|
|Last modified on||2013-03-22 18:33:25|
|Last modified by||gel (22282)|