Most properties on the uniform law of large numbers impose that parameter lies in a compact set.
Let $X_1, \dots, X_n$ be ergodic variables such that, for any $\theta \in \Theta$, $S_n(\theta) = (1/n)\sum_{i = 1}^n f(X_i, \theta)$ converges in probability to $S_0(\theta) = \lim_{n \to \infty}(1/n)\sum_{i = 1}^n \mathbb{E}(f(X_i, \theta))$, for some measurable function $f$ continuous in $\theta$.
To generalize the convergence to the uniform convergence, we typically need $\Theta$ to be compact and $\mathbb{E}(\sup_{\theta}f(X_i, \theta)) < \infty$. Under these conditions, we can say that $sup_{\theta}\lVert S_n(\theta) - S_0(\theta)\rVert = o_p(1)$.
What if $\Theta$ is not compact (an infinite dimensional set)? For example, Let $u_1, \dots, u_n$ be a nonstochastic sequence, such that $u_i \in \mathcal{N}_i$, where $\mathcal{N}_i$ is a neighborhood subset in $\Theta$. Assume that $T_n (u_1, \dots, u_n) = (1/n)\sum_{i = 1}^n f(X_i, u_i)$ converges in probability to $\lim \mathbb {E}(T_n (u_1, \dots, u_n))$. Intuitively, it is as if I replace $\theta$ with a function. Which conditions should I impose to show that
$$\sup_{u_1\in\mathcal{N}_1,\dots,u_n\in\mathcal{N}_n}\lVert T_n (u_1, \dots, u_n) - {E}(T_n (u_1, \dots, u_n))\rVert = o_p(1)?$$