Question: Suppose $f_n, g_n:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_n\rightarrow 0$ a.e. on $[0,1]$ and $\sup_n\int_{[0,1]}|g_n|dx<\infty$.
- Give an example of two sequences $f_n$ and $g_n$ such that $\int_{[0,1]}f_n g_n$ does not go to $0$ as $n\rightarrow\infty$.
- Prove that for any such sequences $f_n$ and $g_n$, and every $\epsilon>0$, there exists a measurable set $E\subset[0,1]$ such that $m(E)>1-\epsilon$ and $\int_Ef_n g_ndx\rightarrow 0$.
My thoughts: I was thinking of doing something like $f_n=n\chi_{(0,\frac{1}{n}]}$, which I believe would converge pointwise to $1$ a.e…I am just having a hard time trying to think of a $g_n$ that would work such that the integral of their product over $[0,1]$ wouldn't go to $0$….
For the second question, I immediately was thinking Egorov, but I haven't quite been able to figure out how to use it here.
Any suggestions, ideas, etc. are appreciated! Thank you.
Best Answer
Your functions $f_n$ work fine if you take $g_n=1$ for all $n$, since $$ \int_0^1f_n=1 $$ for all $n$.
Given any such pair $\{f_n\}$, $\{g_n\}$, and $\varepsilon>0$, let $k=\sup_n\int_0^1|g_n|$. By Egorov's Theorem there exists $E\subset[0,1]$ with $m(E)>1-\varepsilon$ and $f_n\to0$ uniformly on $E$. . So there exists $n_0$ such that, for all $n>n_0$, we have $|f_n|\leq\varepsilon/k$. Then $$ \int_E|f_ng_n|\leq\frac\varepsilon k\,\int_E|g_n|\leq\varepsilon. $$