Problem on property of sequences of functions in Lp spaces

Question

I was going through the problems from an exam in Measure and Integration Theory and I stumbled upon a problem that stumped me. It is as follows:

Let $(X,\mu)$ be a measure space and let $p_1, p_2 \ge 1, f_n \in L^{p_1}(X)$ and $g_n \in L^{p_2}(X)$ such that $||f_n||_{p_1} \rightarrow 0$ and $||g_n||_{p_2}\rightarrow 0$. Show that $f_n g_n$ converges to $0$ almost everywhere (with respect to $\mu$). Does the statement remain true if, instead of $f_n$ converging in the $L^{p_1}$ norm and $f_n \in L^{p_1}(X)$ we have the condition that $f_n$ converges almost everywhere?

I tried applying Hölder's inequality with various coefficients but to no avail and I'm not sure what else I could try. Am I being really dumb and missing something?

Kavi Rama Murthy · Answer 1 · 2019-02-06T06:26:59.987

0

$\|f_n\|_{p_1} \to 0$ and $\|g_n\|_{p_2} \to 0$ do not imply that $f_ng_n$ tends to $0$ almost everywhere. Unless this part is corrected the question does not make much sense. [There is a standard example of a sequence $f_n$ such that $0 \leq f_n \leq 1$, $\|f_n\|_1 \to 0$ but $f_n$ does not tend to $0$ almost everywhere. Take $g_n=f_n$ to get a counterexample].

edited Feb 06 '19 at 06:26

answered Feb 06 '19 at 06:21

Kavi Rama Murthy

359,332

What would that counterexample be, if you don't mind? Also, would it be true if one or both of the $p_i$ were strictly greater than 1? – anonimnystefy Feb 06 '19 at 12:37
Arrange the characteristic functions of dyadic intervals $(\frac {i-1} {2^{n}},\frac 1 {2^{n}})$ ($1\leq i \leq 2^{n}, n\geq 1$) is a sequence. – Kavi Rama Murthy Feb 06 '19 at 12:39
What about this, though? https://math.stackexchange.com/questions/1281421/convergence-in-lp-implies-convergence-in-measure – anonimnystefy Feb 06 '19 at 13:02
Convergence in measure is different from conveegence almost everywhere. @anonimnystefy – Kavi Rama Murthy Feb 06 '19 at 23:06
Ah, my bad, I meant the former. – anonimnystefy Feb 07 '19 at 00:29

Problem on property of sequences of functions in Lp spaces

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