I am reading through the wiki page of the Gateaux derivative. In an example, they let $\Omega$ be some Lebesgue measurable subset of $\mathbb{R}^n$ and consider the integral operator
$$ E: L_2(\Omega) \to \mathbb{R} $$
given by $$ E(u) = \int_{\Omega} F(u(x))dx $$
where $F:\mathbb{R} \to \mathbb{R}$ is continuously differentiable. They show that the Gateaux derivative of $E$ is given by $$ \psi \mapsto \int_{\Omega} F'(u(x))\psi(x)dx $$
In their last step of the proof, it is (implicitly) claimed that $$ \lim_{\tau \to 0} \int_{\Omega} \int_0^1 F'(u(x)+s\tau \psi(x)) \psi(x) ~ ds ~ dx = \int_{\Omega}F'(u(x))\psi(x)dx $$ Although it makes intuitive sense, I do not see how neither the dominated convergence theorem nor the monotone convergence theorem can be used in this setup, without some further restrictions on $u, \psi$ and $\Omega$. For example, if $\Omega$ is compact and $u$ and $\psi$ are continuous then the result holds since the integrand is bounded over the compact set $\Omega \times [0,1]$, so we can use the DCT. I wonder how to show the result in the more general setting of this wiki-article, where we only assume Lebesgue measurability of $\Omega$ and $u, \psi \in L_2(\Omega)$.
Link to article: https://en.wikipedia.org/wiki/Gateaux_derivative