On One Parameter Groups of Linear Transformations

In connection with the theorem of M. H. Stone2 on the representation of one parameter groups of unitary transformations on Hilbert space J. von Neumann3 has shown (without using any representation theorem) that if a group Ut(X < t < cX) of unitary operators on Hilbert space H is measurable in the sense that (Utx, y) is measurable for each x, y in H, then Ug is necessarily continuous in the strong topology of the ring of bounded operators on Hilbert space, i.e.,