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.