Let $m\in\mathbb{N}; \lambda_m>0; lim_{m\to\infty}\lambda_m = 0; \phi\in C([0, T]); f\in L_2([0, T])$.
Is it possible to exchange the following limit with the integration without further facts?
$$\int_0^T\int_0^T \lim\limits_{M\to\infty}\sum_{m=N}^{M}\lambda_m\phi_m(t)f(t)\phi_m(s)f(s)dsdt$$
I am trying to establish that the above equals $\sum\limits_{m>N}\lambda_m\langle f,\phi_m \rangle^2$, where $\langle ., .\rangle$ is the $L_2$ inner product. However, I am unable to see conditions that would allow use of the (Lebesgue) Dominated or Monotone Convergence Theorems.
Purely for context, I am working through a proof of the Mercer Representation Theorem, and the $\lambda_m$ and $\phi_m$ are eigenvalules and eigenvectors of a compact self-adjoint operator. If the given facts are insufficient to swap the limit and integrals, this will confirms I need to reconsider how I've understood the wider proof.
Many thanks.
PS If someone has a link to a proof of the Mercer Representation Theorem that they're happy with, that would be appreciated also.