I'm trying to formalize the fundamental ideas behind test statistics in the context of hypothesis testing. In particular, I want to define different concepts of test statistics leading to different desireable qualities of tests. I'm having trouble formalizing the idea of a "uniform test statistic" yielding a test of uniformly asymptotic level (defined below) and was hoping for some input.
Setup
Let $(\mathcal{X}, \mathbb{E})$ be a measure space and let $\mathcal{P}$ be a set of probabilty measures on the space. Let furthmore $\mathcal{P}_0 \subseteq \mathcal{P}$. We imagine getting a sample of size $n$ sampled independently, i.e. some $x \in \mathcal{X}^n$ and want to determine whether any of the measures in $\mathcal{P}_0$ are acceptable distributions for the observation.
For each $n \in \mathbb{N}$, we define a test of size $n$ as a partition of $\mathcal{X}^n$ into an acceptance region $\mathcal{A}_n$ and a critical region $\mathcal{A}_n^c$. This partition is also expressed through the critical function $\psi_n: \mathcal{X}^n \to \{0,1\}$ given by
$$ \psi_n(x) = \begin{cases} 0 & \text{if $x \in \mathcal{A}_n$}\\ 1 & \text{if $x \in \mathcal{A}_n^c$} \end{cases} $$
A sequence of tests is either $(\mathcal{A}_n)_{n \in \mathbb{N}}$ or equivalently $(\psi_n)_{n \in \mathbb{N}}$. For a given level $\alpha \in (0,1)$ a sequence of tests is said to have pointwise asymptotic level if
$$ \sup_{\nu \in \mathcal{P}_0} \limsup_{n \to \infty} P_\nu(\psi_n =1) \leq \alpha $$
and uniformly asymptotic level if
$$ \limsup_{n \to \infty} \sup_{\nu \in \mathcal{P}_0} P_\nu(\psi_n =1) \leq \alpha $$
where $P_\nu$ denotes the probability assuming that $X$ has distribution $\nu$.
Pointwise asymptotic test statistic
Let $(T_n)_{n \in \mathbb{N}}$ be a sequence of functions $T_n: \mathcal{X}^n \to \mathbb{R}$. If $T_n(X^n)$ converges in distribution to the same distribution for all $\nu \in \mathcal{P_0}$, we say that $(T_n)_{n \in \mathbb{N}}$ is a pointwise asymptotic test statistic wrt. $\mathcal{P}_0$.
Letting $V$ be a random variable with the same distribution as the limit of $T_n(X)$ for any $\nu \in \mathcal{P}_0$, if $\mathcal{B}$ is a closed set such that $P(V \in \mathcal{B}) \leq \alpha$, we say that the sequence of tests
$$ \psi_n(x) = \begin{cases} 1 & \text{if $T_n(x) \in \mathcal{B}$}\\ 0 & \text{otherwise} \end{cases} $$
is based on the pointwise asymptotic test statistic $(T_n)_{n \in \mathbb{N}}$. Note that it is quite easy to show that this test has pointwise asymptotic level by an application of the Portmanteau theorem (for the second to last inequality) since
$$ \limsup_{n \to \infty} P_\nu(\psi_n = 1) = \limsup_{n \to \infty} P_\nu(T_n(X^n) \in \mathcal{B}) \leq P(V \in \mathcal{B}) \leq \alpha $$
and then taking supremums.
Uniform asymptotic test statistic
What extra assumptions are required to prove uniform asymptotic level of a test constructed as above? We would need to ensure that the $$ \sup_{\nu \in \mathcal{P}_0} P_\nu(\psi_n=1) = \sup_{\nu \in \mathcal{P}_0} P_\nu(T_n(X^n) \in \mathcal{B}) $$
are convergent to $P(V \in \mathcal{B})$. What is a sufficient condition for this to occur?
Hope someone can help!
EDIT: I'm thinking that some conditions are required on $\mathcal{P}_0$, since in the case of $\mathcal{P}$ being all distributions on $\mathbb{R}$ with finite variance and $\mathcal{P}_0$ being all distributions with mean zero, one can show that the usual $t$-test has pointwise asymptotic level but not uniform asymptotic level.