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Given a sequence of functions $g_n$ in $x$ that converges pointwise to some function.

Is $$\lim_{t \to 0} \lim_{n \to \infty} \int g_n(x-t) dx = \lim_{n \to \infty} \lim_{t \to 0} \int g_n(x-t) dx$$? Is this always the case? If not, what conditions are needed to be satisfied first before this is considered true?

EDIT: My original question was wrongly written. This is the question I want to ask instead.

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  • $\begingroup$ When you say that the sequence $g_n$ converges, do you mean pointwise or uniformly? $\endgroup$ – Lior B-S Dec 4 '16 at 11:48
  • $\begingroup$ Perhaps the DCT may be of value? $\endgroup$ – Simply Beautiful Art Dec 4 '16 at 13:08
  • $\begingroup$ I think in this case, no. I am not trying to put the limit inside the integral. $\endgroup$ – user198504 Dec 4 '16 at 13:42
  • $\begingroup$ Related: math.stackexchange.com/q/15240/29024 $\endgroup$ – Tommi Brander Jun 4 '18 at 12:19

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