Action of linear functional on integral depending of parameter

Question

Let $K(x,\omega) \in C^{\infty}(\Omega \times \Omega)$, where $\Omega$ is a domain in $\mathbb{R}^{n}$. Let $\mu$ be a probability measure on $\Omega$. My question is under which conditions an equality $$ L\int\limits_{\Omega} K(\cdot,\omega) \, \mu(d \omega) = \int\limits_{\Omega} LK(\cdot,\omega) \, \mu(d\omega) $$ holds for any $L \in (C^{\infty}(\Omega))^{*}$?