Sections of Sets and Functions
Our definitions today are purely set-theoretical. If is a subset of , then given a point we define the “section” of determined by to be the set
Similarly, we define the section of determined by a point to be the set
Of course, the two concepts are essentially equivalent, and neither really depends on the fact that we have only two factors, but we choose this notation for now. If we don’t so much care about the particular point or , we refer to “an -section” or “a -section”. It should be stressed that these sections are not (as might be supposed) subsets of , but rather an -section is a subset of , while a -section is a subset of .
It should be clear that taking sections commutes with most common set theoretic operations. For example, we compute
Similarly, and ; and similarly for -sections.
Now if is any function defined on and is any point, we define the section of determined by to be the function on defined by . Similarly, the section of determined by a point is the function on defined by . Again, we say that is an -section, and is a -section.
With these definitions down, we turn to measure theory. Let and be measurable spaces, and let be the product space.
If is a measurable rectangle, then every -section is either or , according as or not. Similarly, every -section is either or . That is, every section of a measurable rectangle is measurable. Now we let be the collection of all subsets of for which this is true — if and only if every section of is measurable. Clearly contains all measurable rectangles. It’s also closed under unions and setwise differences — making it a ring — and under monotone limits — making it a -ring. Since is a -ring containing all measurable rectangles, it must contain . Therefore, every section of every measurable set is measurable.
Now if and is any measurable set, then we calculate
Since is measurable, must be measurable, and thus all of its sections are measurable. In particular, is measurable for any measurable , and thus is a measurable function. Similarly we can show that the -section of a measurable function is measurable.
The one caveat is that we treated measurable real-valued functions differently from other ones. Just to be sure, let be a measurable real-valued function, and let be a Borel set. Then we need to ask that be measurable. We can use the above fact that , and the result will follow if we can show that . But we easily calculate
and thus the result follows. The proof that the -section is measurable is similar.