$f_k \rightarrow f$ in $L^p$ and that $g_k \rightarrow g$ weakly in $L^q$. Show that $f_k g_k \rightarrow fg$ weakly in $L^1$ I want to solve the following exercise, and I thankfully welcome some hints. Note that this is not homework.

Problem: Let $1 < p,q < \infty$ be conjugate exponents. Assume $f_k \rightarrow f$ in $(L^p, \| \cdot \|_p)$ and that $g_k \rightarrow g$ weakly in $(L^q, \| \cdot \|_q)$. Show that $f_k g_k \rightarrow fg$ weakly in $(L^1, \| \cdot \|_1)$.

I realize that normed convergence implies weak convergence and that $(L^q)^\ast$ can be identified with $L^p$. What I need to show is that
$$|\phi(f_k g_k) - \phi(f g)| \rightarrow 0, \ \ \forall \phi \in (L^1)^\ast$$
which follows if I can show $\| f_k g_k - f g \|_1 \rightarrow 0$, since
$$|\phi(f_k g_k) - \phi(f g)| \leq \|\phi \| \| f_k g_k - f g \|_1.$$
Thank you in advance!
 A: In the $\sigma$-finite case, we can try this. The dual space of $L^1$ can be identified with $L^\infty$, so what you have to show is that for all $\phi\in L^\infty$, we have 
$$\int_X f_ng_n\phi\to \int_X fg\phi.$$
We have 
$$
\left|\int_X f_ng_n\phi- \int_X fg\phi\right|\leqslant \lVert \phi\rVert_\infty\lVert f-f_n\rVert_p\sup_k\lVert g_k\rVert_q+\left|\int_X fg_n\phi- \int_X fg\phi\right|.
$$
Now use the fact that $\sup_k\lVert g_k\rVert_q$ is finite and $f\phi\in L^p$.

As no assumption is done on the measure space we are working, we have to see how to solve the problem when the measure space is not assumed $\sigma$-finite. 
The functions $\{f_n\}$ and $\{g_n\}$ are fixed. Define for positive integers $n,k$, $I_{n,k}:=\{|f_n|>k^{-1}\}$ and $J_{n,k}:=\{|g_n|>k^{—1}\}$. Each of these sets has finite measure by the assumptions of integrability of powers of these functions, and if we define $X':=\bigcup_{n,k\geqslant 1}I_{n,k}\cup J_{n,k}$ with the underlying measure, we still have a measure space, which is this time $\sigma$-finite. All the integrals in the first part taken on $X$ have the same values if these ones are taken on $X'$.
