Convergence of integrals without the dominated convergence theorem

Question

Let $t>0$ and $\{f_n\}_{n \geq 1} $ be a sequence of functions that do not converge pointwise to any integrable function, but where: \begin{equation} \int_0^t f_n(s) ds \rightarrow F(t) < \infty \end{equation} Let $\{g_n\}_{n \geq 1} $ be another sequence of functions which converges, in $(0,t)$, to a constant $C$ in $L^p$, pointwise, or in some other suitable sense. Can we say anything about the convergence of the sequence \begin{equation} \int_0^t g_n(s) f_n(s) ds \end{equation} given that the dominated convergence theorem is not applicable?

Answer 1

(Since you did not specify the convergence of $\int_0^t f_n dx$, and also used the expression $(0, t)$, I am assuming $t$ to be a constant)

Define $g_n$ by $\frac{Cn}{t}x$ on $(0, \frac{t}{n})$ and $C$ everywhere else. Then $g_n \to C$ both pointwise and in the $L^p$ sense. Define $f_n$ for even integers by $nK$ (where $K$ is a nonzero constant) on $(0, \frac{t}{n})$, and $0$ everywhere else, and for odd integers $K$ everywhere. Then $\int_0^t f_n dx = Kt$ identically for all $n$.

$\int_0^t g_n f_n dx$ for even integers gives us $\frac{KCt}{2}$, but for odd integers we easily see the integral converges to $KCt$.