When is the sup of uncountably many random variables measurable?

Question

Let $g= (g_i)_{i \in \{1, \ldots, n\}}$ be a random vector such that $g_i$ are i.i.d. mean-zero Gaussian random variables with unit variance. Define for any $t \in \mathbb{R}^n$, $$ X_t := \langle g, t \rangle $$ where $\langle g, t \rangle$ denotes the inner product. Let $T \subset \mathbb{R}^n$ be a subset. Is the random variable $$ \sup\limits_{t \in T} X_t $$ measurable for any subset $T \subset \mathbb{R}^n$ (which is not necessarily countable)?

Answer 1

Yes, it is alwasy measurable. This is because any subset of $\mathbb R^{n}$ is separable (w,r.t. the standard metric) and $X_t$ depends continuously on $t$. So $\sup_{t \in T} X_t=\sup_i X_{t_i}$ where $(t_i)$ is a countable dense subset of $T$.