Convergence in $L^2$ norm with $f_n \geq 0$

Asked Apr 08 '19 at 18:09

Active Apr 08 '19 at 18:28

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So, I know there is already a post which pretty much answers the following question:

Given that $f_n \to f$ almost everywhere and $\int |f_n|^2 \to \int |f|^2$, show that $f_n \to f$ in $L^2$ norm.

Of course, it is assumed that $f \in L^2$ and $\{f_n\} \subset L^2.$ However, my task now is to give a proof of the same result which makes use of the additional hypothesis that $f_n \geq 0$ for every $n$.

The only thing that I can immediately think of using is Fatou's Lemma. With the additional hypothesis, we can invoke Fatou to show that $$\int f \leq {\lim \inf} \int f_n$$

However, I'm currently unsure of how I may incorporate this result into a working proof. Would anyone happen to have any suggestions?

edited Apr 08 '19 at 18:18

asked Apr 08 '19 at 18:09

Michael

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See here – user657324 Apr 08 '19 at 18:29
I already made reference to that post, but the proofs given there don't satisfy my requirements – Michael Apr 08 '19 at 18:32
Why do you want to use additional hypotheses? Are you leaving some of the old ones out? – Umberto P. Apr 08 '19 at 20:38

Convergence in $L^2$ norm with $f_n \geq 0$

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