A solution to Haagerup's problem: Normality of weights is commutatively determined

We prove that a weight on a C$^*$-algebra is lower semi-continuous if it is lower semi-continuous on commutative C$^*$-subalgebras. This can be used to prove a von Neumann algebraic counterpart that a weight on a von Neumann algebra is normal if it is normal on commutative von Neumann subalgebras. This provides an affirmative solution to Haagerup's second problem (Problem 1.11) on normal weights posed in 1975.