Measurable Subspaces II
Last time we discussed how to define a measurable subspace of a measurable space in the easy case when is itself a measurable subset of : .
But what if isn’t measurable as a subset of ? To get at this question, we introduce the notion of a “thick” subset. We say that a subset of a measure space is thick if for all measurable . If is itself measurable (as it often is), this condition reduces to asking that . If, further, , then we ask that . As an example, the maximally nonmeasurable set we constructed is thick.
Now I say that if is a thick subset of a measure space , if consists of all intersections of with measurable subsets of , and if is defined by , then is a measure space. This definition of is unambiguous, since if and are two measurable subsets of with , then . The thickness of implies that , and we know that
Since , the second term must be zero, and so . Therefore, , and is indeed unambiguously defined.
Now given a pairwise disjoint sequence of sets in , define to be measurable sets so that . If we define
then we find
and so . Therefore
which shows that is indeed a measure.