$L^p$ convergence of a bounded sequence which converges almost everywhere

Question

I'm having a little trouble with this homework problem:

Suppose $\mu(X)<\infty$, $f_n\in L^1$, $f_n\to f$ a.e., and there exists $p>1$ and a constant $C>0$ such that $$\|f_n\|_p\leq C$$ for all $n.$ Prove that $f_n\to f$ in $L^p$.

Here's what I've done so far.

Without loss of generality, suppose that $\mu(X)=1$. Since $\|f_n\|_p\leq C$, we have that $f_n\in L^p$ for all $n$. Furthermore, since $|f_n|^p\to |f|^p$, we have that $$\int |f|^p\leq \liminf \int|f_n|^p\leq C^p$$ by Fatou's lemma, so $f\in L^p$ as well.

Since $|f_n- f|\to 0$, we have that $|f_n-f|^p\to 0$.

I am thinking of applying the Dominated convergence theorem to $|f_n-f|^p$, but I cannot think of a bound.

Answer 1

I'll expand on the comment by Joey Zou:

This is not true. Just take any sequence of functions which converges to $0$ a.e. and has constant $L^p$ norm. It is true that $f_n\rightarrow f$ in $L^q$ for any $q<p$, however.

Indeed, a concrete counterexample is $f_n(x)=n^{1/p}\chi_{[0,1/n]}$ on $[0,1]$ with the Lebesgue measure.

The convergence in $L^q$ for $q<p$, can be shown as follows:

1. By Egorov's theorem, there is a subset $E\subset X$ of arbitrarily small measure, such that $f_n$ converge uniformly on $X\setminus E$.

2. By Hölder's inequality, $$\left(\int_E |f_n-f|^q\right)^{1/q} \le \mu(E)^{1/q-1/p}\left(\int_E |f_n-f|^p\right)^{1/p}$$ hence $\int_E |f_n-f|^q$ is small when $\mu(E)$ is small.

The detailed proof can be found in If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$