Sequences of Measurable Functions
We let be a sequence of extended real-valued measurable functions on a measurable space , and ask what we can say about limits of this sequence.
First of all, the function is measurable. The preimage is the union of the countable collection , while the preimage is the intersection of the countable collection . And so both of these sets are measurable, and we can restrict to the case of finite-valued functions.
So now let’s use our convenient condition. Given a real number we know that if and only if for some . That is, we can write
Each term on the right is measurable since each is a measurable function, and so the set on the left is measurable. Thus we conclude that is measurable as well.
Similarly, we find that the function is measurable.
Now the functions
are also measurable. Indeed, in proving that is measurable we can use the exact same technique as above to prove that the inner supremum is measurable; it doesn’t really depend on the supremum starting at or higher. And then the outer infimum is exactly as before. Proving is measurable is similar.
Now we can talk about pointwise convergence of a sequence of measurable functions. That is, for a fixed point we have the sequence which has some limit superior and some limit inferior . If these two coincide, then the sequence has a proper limit . But one of our lemmas tells us that the set of points where any two measurable functions coincide has a nice property: has a measurable intersection with every measurable set. And thus if we define the function on this subspace of for which the limit exists, the resulting function is measurable.