I have the exercise.
Suppose $1<p<\infty$ and $p^{-1}+q^{-1}=1$. Let f, g be lebesgue measurable functions on $\mathbb{R}$ and let $g_n$ be a sequence of lebesgue measurable function on $\mathbb{R}$ that satisfie the following three conditions.
(a) $f \geq 0$ and $\int_{\mathbb{R}}(f(x))^pm(dx) < \infty$
(b) $\sup_{n \geq 1} \int_{\mathbb{R}}|g_n(x)|^qm(dx) < \infty$
(c) $\lim_{n \rightarrow \infty}\int_{\mathbb{R}}|g_n(x)-g(x)|m(dx) = 0$
where $m$ is a lebesgue measure on $\mathbb{R}$. Then prove the following problem.
(1) $\sup_{n \geq 1} \int_{\mathbb{R}}f(x)|g_n(x)|m(dx) < \infty$.
(2) $\int_{\mathbb{R}}|g(x)|^qm(dx) < \infty$.
(3) $\lim_{n \rightarrow \infty}\int_{\mathbb{R}}f(x)g_n(x)m(dx) = \int_{\mathbb{R}}f(x)g(x)m(dx) $.
My answer.
(1) Using Holder inquality, $\sup \|fg_n\|_{L^1} \leq \sup \|f\|_{L^p} \|g_n\|_{L^q} \leq \|f\|_{L^p}\sup \|g_n\|_{L^q} < \infty$ (By (a),(b))
(2)By (c), there are subsequence $g_{n_k}$ such that $\lim_{k \rightarrow \infty}g_{n_k}(x) = g(x) \ a.e.$ Using Fatou’s lemma, $\int_{\mathbb{R}}|g(x)|^qm(dx) \leq \liminf_{k \rightarrow \infty}\int_{\mathbb{R}}|g_{n_k}|^qm(dx) \leq \liminf_{k \rightarrow \infty}\sup \int_{\mathbb{R}}|g_{n}|^qm(dx) = \sup \int_{\mathbb{R}}|g_{n}|^qm(dx) < \infty$
I can't solve (3).Thank you in advance.
By what was done before, all the integrals in question (3) make sense. To show the wanted convergence, it suffices to show that $$ I_n:=\int_{\mathbb{R}}f(x)\left\lvert g_n(x)-g(x)\right\rvert m(dx)\to 0. $$ To do so, we split the integral over the set $\{f\gt M\}$ in order to get $$ I_n\leqslant M\int_{\mathbb{R}}\left\lvert g_n(x)-g(x)\right\rvert m(dx) +\int_{\mathbb{R}}f(x)\mathbf{1}_{\{f(x)\gt M\}}\left\lvert g_n(x)-g(x)\right\rvert m(dx). $$ The second term can be controlled by Hölder's inequality applied to the maps $x\mapsto f(x)\mathbf{1}_{\{f(x)\gt M\}}$ and $x\mapsto \left\lvert g_n(x)-g(x)\right\rvert$ and the exponents $p$ and $q$.