Mean Convergence is Complete
I evidently have to avoid saying in post titles because WordPress’ pingbacks can’t handle the unicode character ¹…
Anyhow, today we can show that the norm gives us a complete metric structure on the space of all integrable functions on . But first:
If is a mean Cauchy sequence of integrable simple functions converging in measure to , then it converges in the mean to as well. Indeed, for any fixed the sequence is a mean Cauchy sequence of integrable simple functions converging in measure to . Thus we find
But the fact that is mean Cauchy means that the integral on the right gets arbitrarily small as we take large enough and . And so on the left, the integral gets arbitrarily small as we take a large enough . That is,
and converges in the mean to .
As a quick corollary, given any integrable function and positive number , there is some integrable simple function with . Indeed, since is integrable there must be some sequence of integrable simple functions converging in measure to it, and we can pick for some sufficiently large .
Now, if is a mean Cauchy sequence of integrable functions — any integrable functions — then there is some integrable function to which the sequence converges in the mean. The previous corollary tells us that for every there’s some integrable simple so that . This gives us a new sequence , which is itself mean Cauchy. Indeed, for any choose an large enough so that . Then for we have
Since is Cauchy in measure it converges in measure to some function . Then our first result shows that also converges in the mean to — that goes to zero as goes to . But we can also write
so if we choose a large enough , this will get arbitrarily small as well. That is, also converges in the mean to .
This shows that any mean Cauchy sequence of integrable functions is also mean convergent to some function, and thus the space of integrable functions equipped with the norm is a Banach space.