Approximating Functions Limit Integral

Question

Let $f, g \in L^1[0,1)$ such that $fg\in L^1[0,1)$. Can we find sequences $\{f_k\}\subseteq L^2[0,1)$ and $\{g_k\} \subseteq L^2[0,1)$ such that $$ \lim_{k\to\infty} \int_0^1 f_k(x)g_k(x)\,dx = \int_0^1 f(x)g(x)\,dx? $$ I've tried playing around by choosing $f_k$ and $g_k$ such that $$ \|f_k-f\|_{L^1} \to 0 \quad \& \quad \|g_k-g\|_{L^1} \to 0 $$ as $L^2$ is dense in $L^1$ and trying to find a suitable triangle inequality trick, but alas I haven't been successful. Any advice would be appreciate or a counterexample would be nice too.

Answer 1

Consider the truncated functions

$$ f_k(x) = f(x)\cdot\mathbf{1}_{\{|f|\leq k\}}(x)=\begin{cases}f(x)&|f(x)|\leq k,\\ 0,&\text{otherwise}\end{cases} $$

and likewise $g_k(x) = g(x)\cdot\mathbf{1}_{\{|g|\leq k\}}(x)$. Since each of $f_k$'s and $g_k$'s is bounded, it is in $L^2[0,1)$. Moreover, it is clear that

• $f_k \to f$ and $g_k \to g$ almost everywhere, and
• $|f_k g_k| \leq |fg|$.

So by the dominated convergence theorem, $\int f_k g_k \to \int f g$.