Convergence in $L^p$ and $L^q$

Question

I'm having difficulties to prove the following:

Let $1\leq q < p < \infty$. Suppose $f_n\in L^p(E)$ $\forall n\in\mathbb{N}$, and there exists a finite constant $M>0$ such that $$\|f_n\|_{L^p(E)}\leq M \qquad \forall n\in \mathbb{N}.$$ a) If $\|f_n - f\|_{L^1(E)}\to 0$, prove that $f\in L^p(E)$ and $\|f_n - f\|_{L^q(E)}\to 0$.

b) If $m(E)<\infty$ and $f_n\to f$ a.e. in $E$, show that $f\in L^p(E)$ and $\|f_n-f\|_{L^q(E)}\to 0$.

First, let me clarify that I consider myself a beginner in analysis. I understand the proofs of some inequalities like Hölder's, Minkowski's and Cauchy-Schwarz. However I believe Hölder's cannot be used here because $p,q$ are not conjugates and I think I cannot use Cauchy-Schwarz because the problem is for $L^p$ spaces rather than $L^2$. I'm also aware of some other results like Fatou's Lemma, the dominated convergence theorem and Egorov's theorem and I have used them in previous exercises. Although I know these theorems and inequalities and understand their proofs, I don't see how I can use any of them in this problem. As I already mentioned, I'm a beginner and I believe I lack intuition to use these fundamental results. If you have any advice on how to acquire this 'intuition' I will also appreciated.

${\bf Ideas\,\,for\,\,solution:}$

a) Since $L^1$ is complete and $\|f_n-f\|_{L^1}\to 0$ we can find subsequences $f_{n_k}$ such that $f_{n_k} \to f$ for a.e. $x$. I solved this problem using the advice on the Davide's reply.

b) Since $q<p$, $m(E)<\infty$, and $f_n\in L^p$, then $f_n\in L^q$. Similarly, if I can show that $f\in L^p$ then $f\in L^q$. I believe Egorov's theorem could be useful for this proof (as it was for this other proof ) but I don't know how to. What I tried:

Let $m(F)<\epsilon$ and $f_n\to f$ uniformly in $E\setminus F$ then by Egorov's theorem: \begin{align} \int_E|f_n-f|=&\int_{E\setminus F}|f_n-f|+\int_F |f_n-f|\\ \leq& \epsilon m(E\setminus F)+\int_F |f_n-f|^{1-q}|f_n-f|^q\\ \end{align}

I cannot find a way to proceed.

Thanks.

Answer 1

a) Your idea is good. Now apply Fatou's lemma to the sequence $\left(f_{n_k} \right)$. For the second part, use the pointwise inequalities \begin{align} \left|f_n-f\right|^q&\leqslant R^{q-1}\left|f_n-f\right| +\left|f_n-f\right|^q\mathbf 1\left\{\left|f_n-f\right|\gt R\right\} \\ &=R^{q-1}\left|f_n-f\right| +\left|f_n-f\right|^p\left|f_n-f\right|^{q- p} \mathbf 1\left\{\left|f_n-f\right|\gt R\right\}\\ &\leqslant R^{q-1}\left|f_n-f\right| +\left|f_n-f\right|^pR^{q- p} \\ &\leqslant R^{q-1}\left|f_n-f\right| +2^{p-1}\left(\left|f_n\right|^p+\left|f\right|^p\right)R^{q- p} , \end{align}
then integrate.

b) Indeed, we can use Egoroff's theorem: fix $\varepsilon$ and consider $F$ of measure smaller than $\varepsilon$ and $f_n\to f$ uniformly on $E\setminus F$. Then use Hölder's inequality with the exponent $p/q$ and its conjugate to control $\int_F\left|f_n-f\right|^q$.