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Let $f : X \to Y$, $X \subset R^n$, $Y$ Banach space, $g : X \times Y \to R \cup \{ \infty \}$, $L^n$ the n-dimensional Lebesgue measure.

Are there some results under which the following interchange of supremum and integration $$ \sup_{f:X \to Y} \int_X g(x, f(x)) \mathrm{d}L^n(x) = \int_X \sup_{q \in Y} g(x, q) dL^n(x) $$ is valid?

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    $\begingroup$ My first instinct would be to look for a sequence $f_n$ that approximates the sup on both sides, and then use the monotone convergence theorem. $\endgroup$
    – Nate Eldredge
    Jan 16, 2018 at 15:05
  • $\begingroup$ Your idea sounds reasonable, I will try and see if it works out. $\endgroup$
    – yon
    Jan 16, 2018 at 17:03
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    $\begingroup$ On the right hand side you have a measurability problem. Apart from this, if (almost) all suprema $\sup\limits_{q\in Y}g(x,q)$ are attained then every function $f$ which chooses a maximal point (i.e., $f(x)$ satisfies $g(x,f(x))=\sup\limits_{q\in Y}g(x,q)$) gives equality. $\endgroup$
    – Jochen Wengenroth
    Jan 17, 2018 at 9:36
  • $\begingroup$ One the right-hand side I would guess you should take the $esssup$ or else you can have a pathological version $\endgroup$
    – ABIM
    May 29, 2019 at 21:32
  • $\begingroup$ @JochenWengenroth Could you please kindly expand on your comment, or point to a reference. Thanks in advance. $\endgroup$
    – dohmatob
    Jul 31, 2019 at 13:20

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