$L_1$ convergence, pointwise convergence and the supremum

Question

$(f_n):X\to [0,+\infty)$ is a sequence of integrable functions that converges pointwise to an integrable function $f$, and such that $$\lim_{n\to \infty} \int_X f_ndx= \int_X fdx$$

Is the function $h(x)= \sup\limits_n f_n(x)$ integrable?

Answer 1

The answer is no. Consider the following sequence in $X=[0,\infty)$:

$f_1(x)=1$ on $[0,1]$ and $f_1(x)=0$ otherwise;

$f_2(x)=1$ on $[1,1+1/2]$ and $f_2(x)=0$ otherwise;

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$f_n(x)=1$ on $[\sum_{k=1}^{n-1} 1/k, \sum_{k=1}^{n} 1/k]$, $f_n(x)=0$ otherwise.

Clearly, $f_n(x)\to 0$ pointwise and $\int_Xf_n(x)\,dx=1/n\to 0$ BUT $sup f_n(x)\equiv 1$ on $X$ which is not integrable.