Let be a finite real valued measurable function on a measurable set . Show that the set is measurable.
We define . This is popularly known as the graph of a function. Without loss of generality, we may assume that is nonnegative. This is because we can write , where we split the function into two nonnegative parts.
The proof here can also be found in Wheedon’s Analysis book, Chapter 5.
The strategy for proving this question is to approximate the graph of the function with arbitrarily thin rectangular strips. Let . Define , .
We have , where indicates outer measure.
Also, , where are disjoint.
If , we can conclude and thus is measurable (and has measure zero).
If , we partition into countable union of sets each with finite measure. By the same analysis, each is measurable (and has measure zero). Thus is a countable union of measurable sets and thus is measurable (has measure zero).