I would like a ask whether there is any statistical reference containing the following functional analytic argument for the existence part of Neyman-Pearson:
Let $(R, \mathcal{F}, \mu)$ be a measure space. Consider probability measures $d\nu_0 = f_0 d \mu$ and $d\nu_1 = f_1 d \mu$.
Define $\Phi_{\alpha}$ to be the class of measurable maps $\phi : R \rightarrow [0,1]$ such that $\int \phi d \nu_0 = \alpha$.
Claim There exists $\phi^* \in\Phi_{\alpha}$ such that $\int \phi d \nu_1 = \sup_{\phi \in \Phi_{\alpha}} \int \phi d \nu_1$.
Proof. Take a sequence $\phi_n$ such that $$ \lim_{n \rightarrow \infty} \int \phi_n d \nu_1 = \sup_{\phi \in \Phi_{\alpha}} \int \phi d \nu_1. $$
$\Phi_{\alpha}$ is a bounded set in $L^{\infty}(\mu)$, which is the Banach space dual of $L^1(\mu)$. By Banach-Alaoglu, $\phi_n$ contains a weak-* convergent subsequence $\phi_{n_k}$, i.e. there exists $\phi^* \in L^{\infty}(\mu)$ such that
$$ \int \phi_{n_k} f du \rightarrow \int \phi^* f du. $$
Now $\phi^*$ must take value in $[0,1]$ because each $\phi_{n_k}$ does, and
$$ \int \phi^* f_1 du = \lim_{k \rightarrow \infty} \int \phi_{n_k} f_1 du = \sup_{\phi \in \Phi_{\alpha}} \int \phi d \nu_1. $$
Not a 100% sure answer, but: cursory study makes it look like this argument is given in Appendix 3 ("The weak compactness theorem") of Lehman's Testing Statistical Hypotheses (1959 edition).