Set of measurable function on a Hilbert space

Question

Is the set of Borel measurable function $f:H\to \mathbb{C}$ on a separable Hilbert space $H$ a fisrt-countable space for some appropriate topology, as the topology of convergence in measure?

Answer 1

Yes. Indeed, let $B(H,\Bbb C)$ be the set of all Borel measurable functions from $(H,\mu)$ to $\Bbb C$. For each $f,g\in B(H,\Bbb C)$ put $d_\mu(f,g)=\min\{1, \sup_{\epsilon>0} \mu\{x\in H:|f(x)-g(x)|>\varepsilon\}>\varepsilon\} $. Then $d_\mu$ is a pseudometric on $ B(H,\Bbb C)$ and a sequence $\{f_n\}$ of elements of $B(H,\Bbb C)$ converges to a function $f\in B(H,\Bbb C)$ in measure $\mu$ iff $\lim_{n\to\infty} d_\mu(f,f_n)=0$. Also we can endow even the set $\Bbb C^H$ of all functions from $H$ to $\Bbb C$ with a uniform metric $d_u$ by putting $d_u(f,g)=\min\{1,\sup_{x\in H} |f(x)-g(x)|\}$ for each $f,g\in \Bbb C^H$.