More Measurable Real-Valued Functions
We want a few more convenient definitions of a measurable real-valued function. To begin with: a real-valued function on a measurable space is measurable if and only if for every real number the set is measurable.
Indeed, say is measurable. If we take then is a Borel set and . The measurability of tells us that is measurable as a subset of .
Conversely, suppose that the given sets are all measurable. If are real numbers, then we can write , and thus
That is, if is any semiclosed interval then is the difference of two measurable sets, and is thus measurable itself. If is the collection of all the subsets for which is measurable, then is a -ring containing all semiclosed intervals. It must then contain all Borel sets, and so is measurable.
The same statement will hold true if we replace by , or by , or by . We walk through the exact same method as before, constructing left- or right-semiclosed intervals — and thus all Borel sets — from open or closed rays as needed.
In fact, we can even restrict to lie in some everywhere-dense subset . For example, we might only check this condition for rational . Indeed, say we want to construct the closed interval . By density, we can find sequences (increasing) and (decreasing) of points in converging to and , respectively. Then we can construct the intervals or , and their intersection is the closed interval we want. Then the closed intervals generate the Borel sets, and we’re done.
All of these proofs, by the way, hinge on the fact that taking preimages and intersections commute with all of our set-theoretic constructions.
Now, if is a nonzero constant function , then is measurable if and only if is a measurable subset of itself. Indeed, , and is either or , according as does or does not contain . And since every must be contained in some measurable set, must be measurable for to be measurable.
More generally, the characteristic function of a set is measurable if and only if is a measurable subset of . This time, , and is either or , according as contains or not.
If is a measurable function and is a nonzero real number, then the function is also measurable. Indeed, it’s clear that . We must check that is measurable, but this set is equal to , which is measurable.
Finally, every continuous function is Borel measurable. Indeed, we can write any Borel set as a limit of open sets. The preimage of each open set is open, and thus Borel, and the preimage of the limit is the limit of the preimages, which is again Borel.