Relationship between L^p convergence and a.e. convergence

It turns out that convergence in Lp implies that the norms converge. Conversely, a.e. convergence and the fact that norms converge implies Lp convergence. Amazing!

Relationship between $L^p$ convergence and a.e. convergence:
Let $f, \{f_k\}\in L^p$ , $0<p\leq\infty$ . If $\|f-f_k\|_p\to 0$ , then $\|f_k\|_p\to\|f\|_p$ . Conversely, if $f_k\to f$ a.e.\ and $\|f_k\|_p\to\|f\|_p$ , $0<p<\infty$ , then $\|f-f_k\|_p\to 0$ . Note that the converse may fail for $p=\infty$ .

Proof:
Assume $\|f-f_k\|_p\to 0$ .

(Case: $0<p<1$ ).
Lemma 1:
If $0<p<1$ , $|a+b|^p\leq|a|^p+|b|^p$ for all $a,b\in\mathbb{R}$ .
Proof of Lemma 1:
$\displaystyle 1=\frac{|a|}{|a|+|b|}+\frac{|b|}{|a|+|b|}\leq\left(\frac{|a|}{|a|+|b|}\right)^p+\left(\frac{|b|}{|a|+|b|}\right)^p=\frac{|a|^p+|b|^p}{(|a|+|b|)^p}.$
Hence $|a+b|^p\leq(|a|+|b|)^p\leq|a|^p+|b|^p$ .
End Proof of Lemma 1.
Hence, using $|a|^p\leq|a-b|^p+|b|^p$ and $|b|^p\leq|a-b|^p+|a|^p$ we see that $\displaystyle ||a|^p-|b|^p|\leq|a-b|^p.$

Thus
$\begin{aligned} \left|\|f_k\|_p^p-\|f\|_p^p\right|&=\left|\int(|f_k|^p-|f|^p)\right|\\ &\leq\int\left||f_k|^p-|f|^p\right|\\ &\leq\int|f_k-f|^p\\ &=\|f-f_k\|_p^p\to 0\ \ \ \text{as}\ k\to\infty. \end{aligned}$

Hence $\|f_k\|_p\to\|f\|_p$ .

(Case: $1\leq p\leq\infty$ .)

By Minkowski’s inequality, $\|f\|_p\leq\|f-f_k\|_p+\|f_k\|_p$ and $\|f_k\|_p\leq\|f-f_k\|_p+\|f\|_p$ so that $\displaystyle \left|\|f_k\|_p-\|f\|_p\right|\leq\|f-f_k\|_p\to 0$ as $k\to\infty$ . Done.

Converse:

Assume $f_k\to f$ a.e.\ and $\|f_k\|_p\to\|f\|_p$ , $0<p<\infty$ .
Lemma 2:
For $a,b\in\mathbb{R}$ , $|a+b|^p\leq 2^{p-1}(|a|^p+|b|^p)$ for $1\leq p<\infty$ .
Proof of Lemma 2:
By convexity of $|x|^p$ for $1\leq p<\infty$ , $\displaystyle \left|\frac 12 a+\frac 12 b\right|^p\leq\frac 12 |a|^p+\frac 12 |b|^p.$
Multiplying throughout by $2^p$ gives $\displaystyle |a+b|^p\leq 2^{p-1}(|a|^p+|b|^p).$

Thus together with Lemma 1, for $0<p<\infty$ we have $|f-f_k|^p\leq c(|f|^p+|f_k|^p)$ with $c=\max\{2^{p-1}, 1\}$ .

Note that $|f-f_k|^p\to 0$ a.e.\ and $\phi_k:=c(|f|^p+|f_k|^p)\to\phi:=2c|f|^p$ a.e.\ which is integrable. Also, $\int\phi_k\to\int\phi$ since $\|f_k\|_p^p\to\|f\|_p^p$ . By Generalized Lebesgue’s DCT, we have $\int |f-f_k|^p\to 0$ thus $\displaystyle \|f-f_k\|_p\to 0.$

(Show that the converse may fail for $p=\infty$ ):

Consider $f_k=\chi_{[-k,k]}\in L^\infty(\mathbb{R})$ . Then $f_k\to f$ a.e.\ where $f(x)\equiv 1$ , and $\|f_k\|_\infty\to\|f\|_\infty=1$ . However $\|f-f_k\|_\infty=1\not\to 0$ .