Let $E$ be a spectral measure for $(\Omega, A, H)$ (H is a Hilbertspace and A a $\sigma$-algebra), let g, h $\in H$ be fix. then $E_{g,h}$: A -> $\mathbb{C}$ which is defined as $E_{g,h}(\Delta)$ := $(E(\Delta)g,h)$ is a complex measure. ($\Delta \in A$)
My question might seem weird but what exactly is the construct $(E(\Delta)g,h)$? Is this an integral? If yes, how can I write it as an integral?
If $E$ is a spectral measure for $(\Omega, A,H)$ then it is a map $E: A \to B(H)$ such that for each $\Delta \in A$ we have $E(\Delta)$ is a projection. So for $g,h \in H$, $E(\Delta)g\in H$ and thus $(E(\Delta) g, h) \in \mathbb{C}$ is just a number. Here $(\cdot, \cdot)$ denotes the inner product on the Hilbert space. It so happens that $\Delta \mapsto (E(\Delta) g, h)$ is a complex measure.