This is a question about Section 12.20 on p.318 of Functional Analysis by W. Rudin. The context is projection-valued measures in spectral theory.
I think the following definitions would be direct generalizations to vector measures: Let $\Delta$ be a set (maximal ideal space of a C* algebra of bounded operators on a Hilbert space here), $\mathfrak{M}$ a $\sigma$-algebra on $\Delta$ (Borel algebra here) and $E$ a vector measure on $(\Delta,\mathfrak{M})$ (resolution of identity here). A set $N\subset \Delta$ is null if $E(M)=0$ for each $M\in\mathfrak{M}$ with $M\subset N$. We then identify functions that agree a.e. when defining $L^\infty(E)$. Is this definition equivalent to Rudin's? If so, how do I show this? (I tried but failed... In particular, I don't see how this has to do with open sets.)