Let $K\in L^2([0,T]^2)$, and for each $t\in [0,T]$, let $\mathcal{T}_t \in \mathcal{L}(L^2([0,T]))$ be such that for all $f\in L^2([0,T])$, $\mathcal{T}_t f(s)=\int_0^T K(s,t)K(u,t)f(u)\, d u$. One can show by the integrability of $K$ that $\mathcal{T}_t $ is well-defined. Is it possible to prove that the map $[0,T]\ni t\mapsto \mathcal{T}_t\in \mathcal{L}(L^2([0,T]))$ is strongly measurable?