Mean Convergence Properties of Integrals
First off, the basic definitions of the norm carries across: we set
and we say that a sequence of integrable functions is Cauchy in the mean or is mean Cauchy if goes to zero for sufficiently large and . We immediately find that any sequence that is Cauchy in the mean is also Cauchy in measure, since our earlier proof didn’t really depend on the functions being simple.
We also have a couple more results whose proofs don’t really depend on the simplicity of , and so these carry across without change. If is mean Cauchy, with indefinite integrals , then the collection of set functions is uniformly absolutely continuous. Further, we can define the limiting set function
for every measurable , and we find that this set function is finite-valued and countably additive.
So now we can formally define the obvious notion: the sequence of integrable functions “converges in the mean” or is “mean convergent” to an integrable function if goes to zero as goes to . And, as we might expect, if converges in the mean to then it converges in measure as well.
Indeed, for every we define
and then we see that
Thus must go to as , and so converges in measure to .