Is the set of equivalent classes of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?
2026-03-27 07:50:39.1774597839
Is the set of monotone functions $f:[a,b] \to [0,1]$ compact in $L^2([a,b])$?
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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 $L^2$.