Is the set of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?

Question

Is the set of equivalent classes of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?

For clarity you should say equivalence classes of monotone functions. — Kavi Rama Murthy, Jul 16 '20 at 12:22
@KaviRamaMurthy. for closeness : https://math.stackexchange.com/a/1570240/795522 — user15958, Jul 16 '20 at 12:36

score 2 · Accepted Answer · answered Jul 16 '20 at 12:59

Yes. Let $f_n$ be a sequence of such functions. By the Helly selection theorem, there is a subsequence $f_{n_k}$ converging pointwise to some $f$, which is clearly again monotone. And by dominated convergence this subsequence also converges in $L^2$.

Is the set of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?

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