Does $f_n \rightarrow f$ $\mu$-a.e. and $\lim \int f_n \rightarrow \int f$ imply $f_n \rightarrow f$ in $L^1(\mu)$?

Question

The full question from this practice qual:

True/False: Let $(X,M,\mu)$ be any measure space. If $f_n,f \in L^1(\mu)$ are measurable functions, $f_n \rightarrow f$ $\mu$-a.e. and $\lim \int f_n \rightarrow \int f$, then $f_n \rightarrow f$ in $L^1(\mu)$.

I recognize that the question is essentially asking if I can move the limit inside the integral. The three theorems I have which allow me to do this are: Monotone Convergence Theorem, Fatou's Lemma, and Dominated Convergence Theorem. Since I don't have any increasing sequences of functions or dominating function, I would think if the answer is true, I need to use Fatou's Lemma. But I don't see any way to use Fatou's Lemma to justify it is true (the fact the question says "any" measure space signals to me it may be false, but I've been unable to construct a counter example)

Answer 1

False.

Take $f = 0$.

$f_n = \frac{1}{n}\chi_{[0, n]} -\frac {1}{n} \chi_{[-n, 0)}$.

$f_n \to f$ as $n \to \infty$ a.e.

But $\int |f_n| = 2$. So $f_n$ does not converge to $f$ in $L^1$.

Answer 2

For completeness sake: it is true if you assume that $|f_n|_1\to |f|_1$. Take $g_n = (|f|+|f_n|)-|f_n-f|$. By hypothesis $g_n\to 2|f|$ almost everywhere and $g_n\geqslant 0$, so Fatou says...?

This holds in general for $L^p$ spaces: if $f_n\to f$ almost everywhere and $|f_n|_p \to |f|_p$, then $|f-f_n|_p\to 0$. Again, taking $g_n= 2^{p-1}(|f_n|^p+|f|^p) - |f_n-f|^p$ gives by Fatou the result.

This is one of many results due to Riesz.

In particular, it is true if $f_n$ is positive almost everywhere.

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Note the above gives a very easy proof the simple functions are dense in $L^p$ for any finite $p$: just manufacture a sequence of simple functions that tend pointwise and in norm to your function, which is easy!