Due May 4
Let X be a set and a nonempty collection of subsets of X closed under complements and countable unions of disjoint sets. Then is a -algebra.
Let be a measure space. Its completion is defined as the collection of sets of the form , where and for some with and
- is the smallest -algebra containing and all subsets of its elements of measure 0.
- The function is a complete measure on .
Consider the Lebesgue exterior measure. Then a set is Caratheodory measurable if and only if is Lebesgue measurable.
Let be a measure space, subsets of X such that , and . Then .