Free Career Quiz: Please help to do!
Just came across this neat beginner’s Lebesgue Theory question. As students of analysis know, just to show a set is measurable is no easy feat. The usual way is to use the Caratheodory definition, where a set E is said to be measurable if for any set A, . This can be quite tedious.
Question: Suppose E is a Lebesgue measurable set and let F be any subset of such that (Symmetric Difference is Zero). Show that F is measurable.
The short way to do this is to note that implies , and . This in turn (using a lemma that any set with outer measure zero is measurable) implies the measurability of and .
Next comes the critical observation: . Using the fact that the collection of measurable sets is a -algebra, we can conclude is measurable.
Thus is the union of two measurable sets and thus is measurable.