Let $(\mathbb{R}^k, \mathcal{L}(\mathbb{R}^k), P)$ be the probability space induced by the $k\times 1$ vector valued random variable $X$, with $\mathcal{L}(\mathbb{R}^k)$ the completion of $\mathcal{B}(\mathbb{R}^k)$.
Is it true that $P$ is regular, that is, for any measurable set $A$,
(1) $P(A) = \sup \{ P (C) \mid C \subseteq A, C \text{ compact and measurable} \}$
and
(2) $P(A) = \inf \{ P (C) \mid A \subseteq C, C \text{ open and measurable} \}$
Because $X$ is a random variable, it is bounded in probability, so I can prove that (1) works for $C$ bounded but not necessary closed.
However, I have no idea how to prove that $C$ can be closed in (1) or open in (2).
Yes. Every finite Borel measure on a metric space is regular. Since $P$ is a Borel probability measure and $\mathbb{R}^k$ is a metric space, $P$ is regular.
More details here: https://mathoverflow.net/questions/22174/regular-borel-measures-on-metric-spaces