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!

`$x=\sup\{H(f)\}$`

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. $\endgroup$ – Joseph Van Name Mar 7 '13 at 4:23