In the previous chapters, Bartle showed that that if f is in M(X,X), then the functions are also in M(X,X).
The case of the measurability of the product fg when f, g belong to M(X,X) is a little bit more tricky. If , let be the “truncation of f” defined by
Let be defined similarly. We will work out the proof that and are measurable (Bartle left it as Exercise 2.K).
Each is a function on to .
All of the above sets are in X.
Thus, we may use an earlier Lemma 2.6 to show that the product is measurable.
We also have , and using an earlier corollary that says that if a sequence is in M(X,X) converges to f on X, then f is also in M(X,X), we have that belongs to M(X,X).
Finally, (fg)(x)=f(x)g(x)=, and hence fg also belongs to M(X,X).
This is a very powerful result of Lebesgue integration, since we can see that the theory includes extended real-valued functions, and prepares us to integrate functions that can reach infinite values!