Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$.

Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \exists j\in J\}$$

One may check that $E$ is an open set in the strong operator topology but not in the weak operator topology.

**Question1:** I feel $E$ is not in the sigma algebra generated by the weak operator topology but have no evidence to prove it.

**Question2:** Let us assume that dimension of $H$ is of $c$. It seems that $E$ is WOT measurable if and only if every SOT open set is WOT measurable.