I am pondering on the measurability (implicitly?) required for the stochastic equicontinuity and wondering if the following understanding of mine is valid.
Let
$\Theta$: normed metric space.
$\{H_{n}(\theta ):n\geq 1\}$: a family of random functions defined from
$\Theta \rightarrow \mathbb{R}$
Wikipedia states
$\{H_{n}\}$ is stochastically equicontinuous if, for every $\epsilon > 0$ and ${\displaystyle \eta >0}$, there is a $\delta > 0$ such that:
${\displaystyle \limsup _{n\rightarrow \infty }\Pr \left(\sup _{\theta \in \Theta }\;\sup _{\theta '\in B(\theta ,\delta )}|H_{n}(\theta')-H_{n}(\theta )|>\epsilon \right)<\eta .}$
Here $B(\theta, \delta)$ represents a ball in the parameter space, centred at $\theta$ and whose radius depends on $\delta$.
Let $(\Omega,\mathscr{F},\Pr)$ be the probability space on which $\{H_{n}(\theta ):n\geq 1\}$ are defined. The above condition seems to implicitly requre $$ \Big\{ \omega\;\big|\;\sup _{\theta \in \Theta }\;\sup _{\theta '\in B(\theta ,\delta )}|H_{n}(\omega,\theta')-H_{n}(\omega,\theta )|>\epsilon \Big\}\in \mathscr{F},\tag{*} $$ even though it involves with taking supremum over a set of possibly uncountable cardinality.
The cited paper, Newey, Whitney K. (1991) "Uniform Convergence in Probability and Stochastic Equicontinuity", Econometrica, 59 (4), 1161–1167 JSTOR 2938179, considers the outer measure:
To avoid measurability complications it will be assumed that probability statements such as this are for outer probability.
which is defined on $2^{\Omega}$, so indeed the measurability problem does not occur.
But to check the stochastic equicontinuity condition as is, it does not make sense unless we establish (*). (This occurs when, e.g., 1. $\Theta$ is countable, 2. $H_n$ is continuous in $\theta$ for every $\omega$ and a countable set is dense in $\Theta$.)
Am I correct?