I’ll first talk a bit about the monotone convergence theorem (the one associated with Fatou’s lemma and the dominated convergence theorem). I learned those in undergraduate but never really understood them, by which I mean I was never able to learn them in such a way that I would be more or less guaranteed to be able to reconstruct them. That was a time when I, unlike now, was not aware that simple functions, the completion of which results in functions in space is analogous to the density of rationals in the reals.
The single variable monotone convergence theorem is quite simple and straightforward and easy to prove. Lebesgue’s monotone convergence theorem is analogous to it, but in the much more complex vector space of functions. It turns out that this vector space is a Banach space, meaning that it has a norm with respect to which it is complete.
For real vectors of dimension , as far as monotone convergence is concerned, each of the components one can pick an such that guarantees distance less than . Taking the max of the results in the one-norm of the difference’s being less than . This extends to all -norms for , since iff .
In , we proved Holder’s inequality but not for the case with in in which the case the dual space is . The infinity norm of a function exists iff every -norm exists with .
In case of a finite dimensional vector wherein we assume WLOG that , we have
The limit of both the lower and upper bounds as is , which means by the squeeze theorem,
We can generalize this and give each a positive weight , in which case the in the upper bound of would be replaced by , which does not change the infinity norm. As for Holder’s inequality for finite dimensional vectors in the case of , we have the following trivial proposition:
Proposition 1 For any two dimensional vectors with non-negative components, ,
In order for for any , assuming that it is measurable, it is necessary that . If not, there exists an and an increasing sequence of natural numbers such that for all , which implies that for all
which violates finiteness of .
Assume that is also non-negative. Let be the simple function that takes the value iff , in which case
which can be proved using the monotone convergence theorem.
In general for , we let be a derived sequence of simple functions one which takes the value iff , in which case
We moreover stipulate that if , then . In this case takes on no more than distinct values and
We wish to prove that
In order to do so, we first prove the following.
Proposition 2 If , then . Analogous holds for weighted finite dimensional or countably infinitely dimensional vectors.
Proof: We first prove the discrete case. We take a finite dimensional vector with positive weights in which case the -norm is given by
We can thus WLOG assume the weights to all be . Showing that the derivative of the logarithm of this wrt is non-positive suffices. Formally, we wish to show
Assume WLOG that . Then by Holder’s inequality, on , with the -norm on the first, the -norm on the second
which completes our proof for the finite dimensional case.
In the countable case, we can assume because otherwise all the norms would be infinite. Then, it suffices to take a limit in .
For the measurable function case, the RHS of has as the s and as the weight. As we have an countably infinite dimensional vector with a finite one-norm. For finite vectors we’ve already shown that if , the -norm cannot exceed the -norm. Taking the limit yields
which by completes our proof for all except the infinity norm.
For the infinity norm, by , we simply have
Since from below pointwise, . By definition,
It is obvious that
since is a monotonically increasing sequence of functions. Moreover, if , then there must be some such that . Let be such that monotonically. By definition of the s, we must have . Thus, . That for all implies
which completes our proof.