I realize this question has been asked way too many times, for instance, it's this exact problem which I will put below for convivence:
Let $(X,\Omega, \mu)$ be a $\sigma$-finite measure space with $p \in [1,\infty)$. Suppose that $k: X \times X \longrightarrow \mathbb{C}$ is $\Omega \otimes \Omega$-measurable such that for every $f \in L^p(\mu)$, we have $k(x,\cdot)f(\cdot) \in L^1(\mu)$ for $\mu$-a.e $x \in X$. Moreover suppose that $$x \mapsto \int_{X} k(x,y)f(y) d\mu(y) \in L^p(\mu).$$ Prove that $K: L^p(\mu) \longrightarrow L^p(\mu)$ given by $$Kf(x) = \int_{X} k(x,y)f(y) d\mu(y)$$ is bounded.
I however haven't been able to figure out the problem, even after following the solution outlined in the hyperlink. I followed the same approach of wanting to appeal to the Closed Graph theorem. We let $(f_n,Kf_n) \longrightarrow (0,g) \in L^p(\mu) \times L^p(\mu)$ and we wish to show that $g=0$.
It's not too bad to show that for $\mu$ a.e $x \in X$ we have $y \mapsto k(x,y) \in L^q(\mu)$ which allows you to use Holder's inequality. In particular since $Kf_n \longrightarrow g$ in $L^p(\mu)$, if we show $Kf_n \longrightarrow 0 \in L^p(\mu)$ then we are finished. To this end, we observe that: $$\lVert Kf_n \rVert^p_p = \int_{X} \left| \int_{X} k(x,y)f_n(y) d\mu(y) \right|^p d\mu(x) \leq \int_{X} \lVert k(x,\cdot)f_n \rVert^p_1 d\mu(x)$$ $$\leq \lVert f_n \rVert^p_p \int_{X} \lVert k(x,\cdot) \rVert^p_q d\mu(x).$$ Now $\lVert f_n \rVert^p_p \longrightarrow 0$ so I just need to prove that $\int_{X} \lVert k(x,\cdot) \rVert^p_q d\mu(x) < \infty$, which amounts to proving that $x \mapsto \lVert k(x,\cdot) \rVert_q \in L^p(\mu)$. However I don't see how this is true.
I'm also aware that I may assume $Kf_n \longrightarrow g$ pointwise $\mu$ a.e, but I wasn't able to put this to any use, such as using the dominated convergence theorem for example.
Any help on how to finish this problem would be great.
