Is pointwise limit of random variable is random variable?
Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a complete measure space and $V$ is a real separable Hilbert space with $\mathcal{B}(V)$ as its Borel $\sigma$-algebra.
For each $n\in\mathbb{N}$, let $f_n:\Omega\to V$ be a sequence of random variables. Suppose that there exists $f:\Omega\to V$ such that $$\lim_{n\to \infty} f_n(\omega) = f(\omega)$$ for almost all $\omega$, where the limit w.r.t induced norm $\Vert v \Vert =\sqrt{\langle v, v \rangle} $ in $V$.
Is $f$ a random variable?