Measures on projections in a $W^*$-algebra of type $I_2$

It is shown that for every measure $m$ on projections in a $W^*$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2= m(p)$ for every projection $p$. With regard to J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803-806) it means that the assertion is valid for an arbitrary $W^*$-algebra.