# Which topology for compactness and continuity?

I know just enough functional analysis to know that this is probably a simple question, but not enough to actually know the answer.

Let $A$ be the set of nondecreasing functions $f: [a,b] \to [a,b]$. I know from Helly's selection theorem that every sequence $f_n \in A$ has a subsequence that converges almost everywhere to some $f \in A$.

I also have a real valued function $H$ on $A$. This function is not linear, but I can show that if $f_n \in A$ converges almost everywhere to $f \in A$, then $H(f_n)$ converges to $H(f)$.

I would like a short, simple proof that the function $H$ achieves its maximum on $A$. The clear route to take is to show that $A$ is compact and $H$ is continuous, and I think I have all the ingredients for this, but I am unclear about which topology to choose to make this all work.

Thanks!

• Take the weakest topology on $A$ that makes $H$ continuous, then prove that $A$ must be compact. You can't do any better than that. Note that H(A) is compact. The rest follows. – Andrew Stout Mar 7 '13 at 3:55
• Can't you just take $x=\sup\{H(f)\}$ and $x_{n}$ an increasing function with $x_{n}\rightarrow x$. Then there is a sequence $f_{n}$ of functions in $A$ with $f_{n}\rightarrow f$ and $H(f_{n})>x_{n}$ for all $n$. Then there is a subsequence $f_{n_{k}}$ converging to some $f$, so $H(f_{n})\rightarrow H(f)$, thus $H(f)=x$ and the supremum is reached. You don't need to even mess with topology here. – Joseph Van Name Mar 7 '13 at 4:23

## 2 Answers

It seems to me that you don't want to define $H$ on $A$ at all. Rather, you want to define it on $A/\sim$ where $f\sim g$ if $f=g$ almost everywhere. Give $A$ the uniform topology and $A/\sim$ the quotient topology. Then, the two conditions that you state are precisely what you want. Namely, that $\bar H:A/\sim\to \mathbb{R}$ is continuous because it preserves sequences and $A/\sim$ is compact because every sequence has a convergent subsequence.

A slightly more classical answer than Kieth's answer.

The set of non-decreasing (monotone) functions (equivalence classes) $M$ of functions from $[0,1]$ to $[0,1]$ is norm-compact in the Banach space $L_1$. (Helly's selection theorem)

Now take the function $H$ and suppose that $f,g\in A$ differ only on a Lebesgue-null set. Clearly, the constant sequence $f$ converges to the $g$ almost surely. Thus, $H(f) = H(g)$.

This implies that the function $\tilde H\colon M\to \mathbb{R}$ defined by $\tilde H([f]) =H(f)$, for each equivalence class $[f]\in M$, $f\in A$, is well defined. It is also continuous.