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Suppose I have a double sequence of bounded $L^{p}$ functions $f_{n,m}$ defined on some open set of $\mathbb{R}^{n}$ such that $f_{.,m}$ converges uniformly for each $m$, such that $f_{n,.}$ converges in $L^{p}$ norm for each $n$, and such that $\lim_{n\rightarrow +\infty}\lim_{m\rightarrow +\infty}f_{n,m}$ and $\lim_{m\rightarrow +\infty}\lim_{n\rightarrow +\infty}f_{n,m}$ exists, are these two limits equal ?

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  • $\begingroup$ You say "bounded L^p functions". Do you mean that the set $\{f_{m,n}\}$ is bounded in L^p? $\endgroup$ – Thompson Apr 25 '17 at 23:11
  • $\begingroup$ @Thompson no I mean bounded in $\mid .\mid_{\infty}$ sense $\endgroup$ – user1611830 Apr 25 '17 at 23:12
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My guess is "no".

The details seem like red herrings. The standard counterexample to limit swapping is the sequence of numbers $m/(m+n)$. Doesn't this work? (Ie constant functions)

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  • $\begingroup$ I said constant functions, I.e. $f_{m,n}(x) \equiv m/(m+n)$ $\endgroup$ – Thompson Apr 25 '17 at 23:19
  • $\begingroup$ To be clear, my aim is to provide a counterexample. $\endgroup$ – Thompson Apr 25 '17 at 23:20
  • $\begingroup$ ok get it. I am going wrong, the supplementary hypothesis is that the convergence of $f_{.,m}$ should be uniform with respect to $m$ $\endgroup$ – user1611830 Apr 25 '17 at 23:32

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