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.
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Hi, thank you for the post. I’m studying real analysis and I find you blog through a google search!
I have a quick question about the first equation. How can I get from the convergence in measure to the convergence of the integral?
We know that |f_n-f_m| converge in measure to |f-f_m|, we have |f_n-f_m| is a mean Cauchy sequence and |f_n – f_m| are simple functions. But I have trouble putting the conditions together to get the convergence of the integral…
It seems that it repeated the conditions for the corollary to be proofed.