Let $ \tau $ be a faithful normal semifinite trace on a von Neumann algebra $ \mathcal{M} $ .We study the cases when a hyponormal $ \tau $ -measurable operator (or a restriction of it)is normal. We obtain a criterion for the hyponormality of a $ \tau $ -measurable operatorin terms of its singular value function. The set of all $ \tau $ -measurable hyponormaloperators is closed in the topology of $ \tau $ -local convergence in measure. This assertionis a generalization of Problem 226 from the book“Halmos P.R., A Hilbert Space Problem Book, Second edition, Springer, New York (1982)”to the setting of unbounded operators.The set of all $ \tau $ -measurable cohyponormal operators is closed in the topology of $ \tau $ -local convergencein measure if and only if the von Neumann algebra $ \mathcal{M} $ is finite.
