integration - If $mu(|f_n|^p)$ is bounded and $f_nto f$ in measure then $f_nto f$ in $L^1$

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

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

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