A strongly open set which is not measurable in the weak operator topology

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

Let us put E={xB(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

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