Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a proper subset in $I$.
Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: j\in J\}$$
One may check that $E$ is an open set in the strong operator topology but not in the weak operator topology.
Question: I feel $E$ is not in the sigma algebra generated by the weak operator topology but have no evidence to prove it.