Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing that $\forall s\in[a,b]\quad|\psi(s)|^2\le\|\varphi\|^2\int_{[a,b]}|K(s,t)|^2d\mu_t$, and the fact that $\int_{[a,b]}\int_{[a,b]}|K(s,t)|^2d\mu_td\mu_s$ exists because $K\in L_2([a,b]^2)$.
I think that such majorations to prove summability work if $\psi$ is measurable, but how can we see that it is? Thank you very mcuh!