$f_n \to f$ a.e. and $\| f_n\|_p \to \|f\|_p$. Is $\{f_n\}$ dominated by some $g$?

Question

Let $E\subset \mathcal{M}(\mathbb{R}^n)$ with $m(E)>0$, $\{f_j\}_{j\in \mathbb{N}}\subset \mathcal{L}^p(E)$ and $f\in \mathcal{L}^p(E).$ Let $1\leq p < + \infty$ and suppose that $f_j\to f$ almost everywhere on E and $||f_j||_p\to ||f||_p$. I want to prove the standard fact that $||f_j-f||_p\to 0.$

I know I have the inequality

$$|f-f_n|^p \leq (2\max(|f|,|f_n|))^p = 2^p \max(|f|^p,|f_n|^p) \leq 2^p (|f|^p + |f_n|^p)$$ for any $p>0.$

So, supposing the sequence $\{f_n\}$ is dominated by some $g$, we have the bound:

$$\{h_n=|f-f_n|^p\}\leq 2^{p+1}g^p$$

Since $h_n \to 0$ almost everywhere, applying dominated convergence theorem we get:

$$\text{lim}||f-f_n||_p^p=\text{lim}\int_E|f-f_n|^p=0$$

hence $$\text{lim}||f-f_n||_p=\left(\text{lim}||f-f_n||_p^p\right)^{1/p}=0.$$

Is it correct until now?

Supposing it is correct, one is left to prove that $\{f_n\}$ is indeed bounded.

How to show that $f_j\to f$ almost everywhere on E and $||f_j||_p\to ||f||_p$ imply that ${f_n}$ is dominated so that one can apply dominated convergence?

Note that there already a few questions on this site that address in some way the statement I want to prove. But I would like to know if my own reasoning is correct and moreover none of them addresses my specific questions.

Answer 1

The fact that $f_n \to f$ almost everywhere and $\|f_n \|_p \to \|f\|_p$ does not imply that $f_n$ is dominated by any $L^p$ function $g$. For example, the sequence of functions

$$f_n = n \chi_{[n, n+ n^{-2}]}$$

converges pointwisely and in $L^1$ norm to the zero function $f = 0$. But the sequence is not bounded by any $L^1$ functions $g$: if $g$ is a function so that $g\ge f_n$ for all $n$, then

$$ \| g\|_1 \ge \sum_{k=1}^n \frac 1k$$

for all $n$.

Indeed you do not need the dominated convergence theorem. Just use Fatou's lemma. See here.

Answer 2

There is actually a slightly different form of Dominated Convergence that we can apply provided we already know the limit is in $L^1.$ That is, suppose that $h_n,g_n,h,g\in L^1$ with $h_n\to h$ and $g_n\to g$ almost everywhere, $|h_n|\leq g_n$ and $\int g_n\to \int g,$ then $\int h_n\to \int h.$ In this case we can let $$h_n=|f_n-f|^p,$$ $$h=0,$$ $$g_n=2^p(|f_n|^p+|f|^p),$$ $$g=2^p(|f|^p+|f|^p).$$ Then as you've shown $|h_n|\leq g_n,$ as well we can easily deduce that $g_n\to g$ almost everywhere and $\int g_n\to \int g$, it follows that $$\int h_n=\int |f_n-f|^p=\int h=\int0=0.$$