I would like a simple example of a series of measurable functions $(f_n)$ where the sum of the integrals of $(f_n)$ does not equal the integral of the sum of $(f_n)$.
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If $n\in\mathbb N$, define$$\begin{array}{rccc}g_n\colon&[0,1]&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}2n^2x&\text{ if }x<\frac1{2n}\\2n-2n^2x&\text{ if }x\in\left[\frac1{2n},\frac1n\right]\\0&\text{ otherwise.}\end{cases}\end{array}$$Then $(g_n)_{n\in\mathbb N}$ converges pointwise to the null function, but $\int_0^1g_n(x)\,\mathrm dx=\frac12$, for each $n\in\mathbb N$.
Now, you can convert this example into a series: just take $f_1=g_1$ and, if $n>1$, $f_n=g_n-g_{n-1}$.
José Carlos Santos
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Could you please add some hypothesis which guarentee the change of sum and integral valid – Cloud JR K Feb 05 '20 at 18:16
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@CloudJR Take a look at this question. – José Carlos Santos Feb 05 '20 at 18:18
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Is uniform convergence of the series a sufficient condition to change integral and sum? Btw I checked that question it doesn't have answer for this. – Cloud JR K Feb 05 '20 at 18:23
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1Yes, uniform convergence is enough. – José Carlos Santos Feb 05 '20 at 18:24