If $\mu(|f_n|^p)$ is bounded and $f_n\to f$ in measure then $f_n\to f$ in $L^1$

Question

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of real measurable functions s.t.,

(a) The sequence $\displaystyle(\int |f_n|^p\ \mathsf d\mu)_{n\in\Bbb{N}}$ is bounded.

(b) The sequence $(f_n)_{n\in\Bbb{N}}$ converges in measure to a measurable function $f$.

Use Hölder's inequality to show that $$\displaystyle\lim\limits_{n\to\infty}\int|f_n-f|\ \mathsf d\mu=0.$$

So far I've shown that $f$ is $p$-integrable and that, for all $n\in\Bbb{N}$ and for all real number $\alpha>0$

$$\displaystyle \int |f_n-f|\ \mathsf d\mu\leq \alpha\mu(\Omega)+\int_{ \{|f_n-f|>\alpha\}}|f_n-f|\ \mathsf d\mu$$

I'm trying to use Hölder on the last term of above inequality to show that this part goes to $0$. But I don't know how to do it.

A hint would be more appreciated then the whole answer.

Answer 1

By Holder's inequality, $$\int_{|f_n-f|>\alpha} |f_n - f| d\mu \leq \int_{|f_n-f|>\alpha} |f_n| d\mu + \int_{|f_n-f|>\alpha} |f| d \mu \leq$$ $$ \mu(\{\omega\in\Omega:|f_n(\omega)-f(\omega)|>\alpha\})^{1/q} \left(||f_n||_p+||f||_p\right)$$

As $n\to\infty$, $\mu(\{\omega\in\Omega:|f_n(\omega)-f(\omega)|>\alpha\})\to 0$, and $\left(||f_n||_p+||f||_p\right)<\infty$.

Then $$\mu(\{\omega\in\Omega:|f_n(\omega)-f(\omega)|>\alpha\})^{1/q} \left(||f_n||_p+||f||_p\right)\xrightarrow{n\to\infty}0$$

Answer 2

We prove $\int |f|^p\,d\mu$ is bounded first.

Since $f_n \to f$ in measure, there is a subsequence $f_{n_k}$ converge a.e. to $f$. By Fatou's lemma $$ \int_{\Omega} |f|^p\,d\mu = \int_{\Omega} \lim\limits_{k \to \infty} |f_{n_k}|^p\,d\mu \leq \liminf\limits_{k \to \infty} \int_{\Omega} |f_{n_k}|^p \,d\mu\leqslant\left(\int_{\Omega} |f_n|^p\,d\mu\right)_{n\in\Bbb{N}}< \infty $$