Simple Question Regarding Sequence of Measurable Functions & Integrals

Question

Let $(X,\mathcal{S},\mu)$ be a measure space and let $f_1,f_2,\cdots$ be a sequence of measurable functions from $X$ to $\mathbb{R}$.

My question is if $$ \lim_{k\rightarrow\infty}\int|f_k|d\mu=0 $$ then does it follow that $\lim_{k\rightarrow\infty}|f_k|$ exists? I'm trying to solve a problem and this result would be very convenient.

Thanks!

Am I crazy, or must you have $\lim_{k\to\infty} f_k = 0$, $\mu$-a.e.? — Integrand, Oct 13 '20 at 01:24
There is a logical error here. If $\lim_{k\rightarrow\infty}|f_k|$ does not exist then the integral is not defined. — Matematleta, Oct 13 '20 at 02:09
You’re right. I made a typo, the limit should be outside the integral. — Karambwan, Oct 13 '20 at 02:26

score 1 · Accepted Answer · answered Oct 13 '20 at 03:40

This is saying that the $f_k$ converge to $0$ in $L^1$. It is a result that this implies that there is a subsequence $f_{n_k}$ which converges to 0 almost everywhere, but in general it is not necessary that $f_{n} \to 0$. A standard counterexample is the "typewriter" sequence; see this MSE post: The Typewriter Sequence.

Simple Question Regarding Sequence of Measurable Functions & Integrals

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