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

Let us put E={x∈B(H):‖

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.

**Answer**

**Attribution***Source : Link , Question Author : ABB , Answer Author : Community*