Prove that a pointwise converging sequence, bounded in $L^p$, converge uniformly in $L^q$, using Egorov's theorem

Question

Let $p > 1$, $E$ be a measurable subset of $\mathbb{R}^n$ with finite Lebesgue measure $\mu$, $\{f_n\}_{n\in N}$ be a sequence of measurable functions $f_n : E → \mathbb{R}$, bounded in $L^p(E)$, and pointwise converging to $f : E → \mathbb{R}$.

Prove that $\{f_n\}_{n∈N}$ converges to $f$ in $L^q(E)$ for all $q ∈(1, p)$.

All the hypothesis of Egorov's theorem are satisfied, then $\forallε > 0$ $\exists N_ε ⊆ Ω$ measurable such that $\mu(N_ε) ≤ ε$ and such that the sequence $f_n$ converges uniformly to $f$ in $Ω \setminus N_ε$.

Any hint on how to start the exercise?

Answer 1

Here's a hint on how to start. Use Egorov's theorem to split the domain into two parts, one with $\epsilon$ measure on which convergence is bad, and one with the majority of the measure on which convergence is good. Then bound the total $q$-norm of the difference of the sequence from the limit by bounding the $q$-norm of the difference on each of these sets.