Yesterday, we showed that the graph of a non-negative measurable function is measurable. Today we’ll explore this further. We continue all the same notation as we used yesterday: is a measurable space and is the measurable space of real numbers and Borel sets.
First, if is any measurable subset, and if and are real numbers, then the set is also measurable. The affine transformation sends any measurable set in to another such set, so if is a measurable rectangle, then the transformed set will also be a measurable rectangle. It’s also straightforward to show that the transformation commutes with setwise unions and intersections, and that the assertion is true for . Thus the assertion holds on some -ring , which contains all measurable rectangles. Since measurable rectangles generate the -algebra of all measurable sets on , must contain all measurable sets, and thus the assertion holds for all measurable .
Now — as a partial converse to yesterday’s final result — if is a non-negative function so that (or ) is a measurable set, then is a measurable function. We will show this using an equivalent definition of measurability — that will be measurable if we can show that for every real number the set is measurable. This is clearly true for every nonpositive , and so we must show it for the positive .
For each positive integer we define the set
Our above lemma shows that the first set in the intersection is measurable — as a transformation of , which we assumed to be measurable — and the second set is a measurable rectangle. Thus the intersection is measurable. We take the union of these sets for all :
This is the union of a sequence of measurable sets, and so it is measurable. Taking any we find the -section determined by is the set . And this is thus measurable, since all sections of measurable sets are measurable.
This result is the basis of an alternative characterization of measurable functions. We could have defined a non-negative function to be measurable if its upper (or lower) ordinate set ( is measurable, and extended to general functions by insisting that this hold for both positive and negative parts.
Finally, we can extend our result from last time. If is any measurable function, then its graph is measurable. Indeed, we can take the positive and negative parts and , which are both measurable. Thus all four sets , , , and are measurable. Choosing and we reflect and to and . We then form the unions
The difference between these two sets is the graph of , which is thus measurable.