Problem Statement: Suppose $f_{n}\in L^{p}(\mathbb{R}^{d})$ with $\lVert f_{n}\rVert_{p}\leq M$ ($1\leq p<\infty$). Suppose $f_{n}(x)\rightarrow f(x)$ pointwise almost everywhere.
Show that $f\in L^{p}(\mathbb{R}^{d})$ with $\lVert f\rVert_{p}\leq c_{p}M$. ($c_{p}=2^{p-1}$, so $\lvert x+y\rvert^{p}\leq c_{p}(\lvert x\rvert^{p}+\lvert y\rvert^{p})$, $p\geq 1$).
I am trying to solve this problem but I am not completely clear about what exactly I need to show, and what I can assume.
I was told to first assume that $f$ is bounded, and use Egorov's theorem to show that $\lVert f\rVert_{L^{p}(B_{R})}\leq c_{p}M$ for each $R>0$.
Egorov's Theorem $D\subset \mathbb{R}^{n}$, $m(D)<\infty$ and $f_{n}, f$ are measurable functions over $D$ with $f_{n}(x)\rightarrow f(x)$ almost everywhere. Then $\forall \varepsilon >0$ $\exists E\subset D$ closed with $m(D\setminus E)<\varepsilon$ and $f_{n}\rightarrow f$ uniformly in $E$.
First, with the original assumption "$f_{n}(x)\rightarrow f(x)$ pointwise almost everywhere", does this mean with respect to the $L^{p}$ norm? That is, $\exists E\subset \mathbb{R}^{d}$ with $m(E)=0$ such that $\forall \varepsilon>0$ $\exists N=N(\varepsilon)>0$ such that $n\geq N$ and $x\in \mathbb{R}^{d}\setminus E$ (or $x\in B_{R}\setminus E$) implies $\lVert f_{n}(x)-f(x)\rVert_{p}<\varepsilon$.
Then to show that $f\in L^{P}(B_{R})$. We must show that $\lVert f\rVert_{p}$ exists? That is, show that $\Big(\int_{B_{R}}\lvert f\rvert^{p}\Big)^{1/p}$ exists. Is it enough to say that because $f$ is measurable, then $\lvert f\rvert^{p}$ is measurable, and thus has an integral? But by Egorov's Theorem, $f_{n}\rightarrow f$ uniformly in $B_{R}$, so is it enough to use the fact that $L^{p}$ is a Banach space to conclude that since $f_{n}\rightarrow f$ uniformly implies that $f\in L^{p}(B_{R})$?
Then we have $$\lVert f\rVert_{p}\leq \lVert f_{n}-f\rVert_{p}+\lVert f_{n}\rVert_{p}<\varepsilon +M.$$
But we must show that $\lVert f\rVert_{p}\leq c_{p}M$. Yet, I am not seeing where the $c_{p}$ comes into play, nor why the inequality $\lvert x+y\rvert^{p}\leq c_{p}(\lvert x\rvert^{p}+\lvert y\rvert^{p})$ is useful.
I appreciate any hints to push me in the right direction.
Best Answer
As noted by @zwh, Fatou's Lemma yields \begin{align} \|f\|_p &= \left(\int_{\mathbb R^d} |f|^p\right)^{\frac1p}\\ &\leqslant \liminf_{n\to\infty} \left(\int_{\mathbb R^d} |f_n|^p\right)^{\frac1p}\\ &= \liminf_{n\to\infty}\|f_n\|_p\\ &\leqslant M. \end{align}