Measure theory convergence in Lp.

Question

We assume that $\mu(E)<\infty$. Let $\left(f_n\right)_{n \in \mathbb{N}}$ and $f$ be real measurable functions on $E$, and let $p \in[1, \infty)$. Show that the conditions (i) $f_n \longrightarrow f$, $\mu$ a.e., (ii) there exists a real $r>p$ such that $\sup _{n \in \mathbb{N}} \int\left|f_n\right|^r \mathrm{~d} \mu<\infty$, imply that $f_n \longrightarrow f$ in $L^p$.\ I need some hint on how I can use these ideas because I got scattered and unable to use any idea properly. Seems like using dominated convergence theorem, yet confused about sup thing here.

Answer 1

The following is a detailed sketch.

By Fatou's lemma $f\in L^r$ so that $g_n:=f_n-f \in L^r$ satisfies all our conditions, with pointwise limit 0. Define $M:=\sup_n \int |g_n|^r\, d\mu$.

For a fixed $t>0$, we have, using Hölder's inequality: \begin{align} \int_E |g_n|^p\, d\mu & = \int_{\{ |g_n|\geq t\}} |g_n|^p\, d\mu + \int_{\{ |g_n|<t\}}|g_n|^p\, d\mu\\ & \leq \mu(\{ |g_n|\geq t\})^{(1-p/r)p}\left(\int_{\{ |g_n|\geq t\}} |g_n|^r\, d\mu\right)^{p^2/r} + t^p\mu(E)\\ & \leq M^{p^2/r}\mu(\{ |g_n|\geq t\})^{(1-p/r)p} + t^p\mu(E). \end{align} The first term goes to 0 (for each fixed $t$!) since $g_n\to 0$ in measure. Now just pick $t>0$ such that the right hand side is as small as you need.