Is almost sure limit of Hilbert space valued random variable is also a random variable?

Let $$(\Omega,\mathcal{F},\mathbb{P})$$ complete probability space. Let $$V$$ be a Hilbert space and denote $$\mathcal{B}(V)$$ be the topology generated by the metric $$d_V(x,y)=\Vert x-y \Vert$$ on $$V$$.

If $$\{X_n:\Omega\to V\}$$ sequence of random variables which converges almost surely on $$X$$, i.e., $$\displaystyle\lim_{n\to\infty} X_n(\omega) = X(\omega) \text{ a.s.}$$ Can we say that $$X$$ is a random variable?

I hope this question does make sense..

For real random variables, the following link argues it does hold provided that the space is complete
Measurability of an a.e. pointwise limit of measurable functions.

any insight can be a huge help for student still learning stuffs.

• There are several notions of meaurability for vector values functions. What exactly is your defintion of a random variable with values in $V$? Commented May 29, 2021 at 11:48
• @KaviRamaMurthy $X:\Omega\to V$ measurable provided, for any $A\in \mathcal{B}(V)$, $\{\omega\in\Omega: X(\omega) \in A\} \in \mathcal{F}$ Commented May 29, 2021 at 11:56
• In that case we cannot say that the limit $X$ is measuarble in general. But if $V$ is separable then the cocnlusion holds. Commented May 29, 2021 at 12:00
• @KaviRamaMurthy Thank you Very much this is very Helpful.. would you mind if I ask reference relating to this question. So that I can further study why existence of ONB is crucial in this case. Commented May 29, 2021 at 12:36
• Vector measures by Diestel and Uhl is a good reference. Commented May 29, 2021 at 13:08