I'm having trouble with this problem and I would like your help getting started with it:
Suppose $f_n, g_n$ are Lebesgue measure functions on $\mathbb{R}$, with $f_n, g_n \geq 0\; \forall n \in \mathbb{N}$. Suppose $f_n \rightarrow f$ a.e., $\ g_n \rightarrow g$ a.e.,
$\int{f_n} \rightarrow \int{f} $
and
$\int{g_n} \rightarrow \int{g} $
Prove or disprove: if $\{f_ng_n\}$ is bounded in $L^1$, then
$\int{f_ng_n} \rightarrow \int{fg}$.
I know that the statement is false if you remove the condition that $\{f_ng_n\}$ is bounded in $L^1$, you can use $f_n = g_n = \frac{1}{\sqrt{x}}\chi_{(\frac{1}{n+1},1)}$, both of which converge to $\frac{1}{\sqrt{x}}\chi_{(0,1)}$ which is in $L^1$, where $fg=\frac{1}{x}\chi_{(0,1)}$ is not $L^1$.
But $\int{f_ng_n} = \log{(n+1)}$, so $\{f_ng_n\}$ is not bounded in $L^1$.
Any help with this problem would be appreciated.
Best Answer
How about $f_n= g_n = \sqrt{n} \chi_{[0,1/n)}$? Then pointwise the functions go to zero and the product is bounded in $L^1$, but $\int f_n g_n =1$ and $\int fg =0$.