Convergence of measurable functions by two conditions

Question

I have the following task from my textbook:

Let $\left\{f_{n}\right\}_{n \in \mathbb{N}}$ be a sequence of measurable function on $M$ with $$ f_{n} \rightarrow f \text { a.s., } $$ where $f$ is also a measurable function. Here, a.s. means almost surely (= almost everywhere).

Show: if there exists a nonnegative measurable function $g$ satisfying the following conditions: $$ \left|f_{n}\right| \leq g \text { a.s. for all } n \in \mathbb{N} $$ and $$ \int_{M} g^{p} d \mu<\infty \text { for one } p>0, $$ then $$ \int_{M}\left|f_{n}-f\right|^{p} d \mu \rightarrow 0 \text { for } n \rightarrow \infty . $$

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These notations remind me a little bit of the Hölder inequality.

What I have got:

I thought that we have $$ |f_n|^p \leq g^p a.s. $$ Thus $$ \int_M |f_n|^p d\mu \leq \int_M g^p d\mu \leq infty $$

And as $$ f_n \rightarrow f $$ then the rest follows?

Answer 1

For $p>1,$ the function $x\mapsto x^p$ is convex so $|f_n-f|^p=2^p\left|\frac{f_n}{2}+\frac{(-f)}{2}\right|^p\leq 2^{p-1}(|f_n|^p+|f|^p)$ so $$ 2^{p-1}(|f_n|^p+|f|^p)-|f_n-f|^p\geq 0. $$ In the standard way that dominated convergence theorem is proven, just apply Fatou's lemma.

Answer 2

Only with the help of @zugzug and @DanielWainfleet I was able to complete the proof:

For $p>1,$ the function $x\mapsto x^p$ is convex, so $$ |f_n-f|^p=2^p\left|\frac{f_n}{2}+\frac{ (-f) }{2}\right|^p\leq 2^{p-1} (|f_n|^p+|f|^p) \leq 2^{p-1} (|f_n|^p+|f|f|^p) -|f_n-f|^p\geq 0. $$ Because of the Fatou lemma and the fact that $f_{n} \rightarrow f$ a. e. , we have $$ \begin{aligned} \int_M 2^{p}|f|^{p} d\mu &=\int_M \liminf_{n\rightarrow \infty} \left (2^{p-1}\left (|f|^{p}+\left|f_{n}\right|^{p}\right) -\left|f_{n}-f\right| ^{p}\right) d\mu \\ & \leq \liminf_{n\rightarrow \infty} \int_M 2^{p-1}\left (|f|^{p}+\left|f_{n}\right|^{p}\right) d\mu+\liminf_{n\rightarrow \infty}\left (-\int_M\left|f_{n}-f\right|^{p}d\mu\right) \\ &=\lim_{n\rightarrow \infty} \int_M 2^{p-1}\left (|f|^{p}+\left|f_{n}\right|^{p} \right) d\mu-\limsup_{n\rightarrow \infty} \int_M \left|f_{n}-f\right| {p}d\mu \\ &=\lim_{n\rightarrow \infty} \int_M 2^{p-1} (2|f|^p) d\mu-\limsup_{n\rightarrow \infty} \int_M \left|f_{n}-f\right|^{p}d\mu \\ &=\int_M 2^{p}|f|^{p}d\mu-\limsup_{n\rightarrow \infty} \int_M \left|f_{n}-f\right|^{p}d\mu \end{aligned} $$ where the final equality follows from the theorem of dominated convergence. As now $$ \int_M 2^{p}|f|^{p} d\mu \leq \int_M 2^{p}|f|^{p}d\mu-\limsup_{n\rightarrow \infty} \int_M \left|f_{n}-f\right|^{p}d\mu $$ applies, shall: $$\limsup_{n\rightarrow \infty} \int_M\left|f_{n}-f\right|^{p} \leq 0$$ and the claim follows. $\quad \blacksquare$