I'm following this text
Let $\mathcal{B}(\mathbb{C})$ be the collection of Borel subsets of $\mathbb{C}$, and $P(\mathcal{H})$ the set of projections on the Hilbert space $\mathcal{H}$, denoting by 1 and 0 the projections $p\in P(\mathcal{H})$ which project to $\mathcal{H}$ and the set $\{0\}$, respectively.
A complex espectral measure is a function $E:\mathcal{B}(\mathcal{C})\to P(\mathcal{H})$ such that:
- $E(\emptyset)=0$ and $E(\mathbb{C})=1$,
- If $\{B_n\}_{n\in\mathbb{N}}$ is a collection of disjoint Borel subsets of $\mathbb{C}$, then $E(\bigcup_{n\in\mathbb{N}}B_n)=\sum_{n\in\mathbb{N}}E(B_n)$.
Everything good up to here for me. My problem is that all of a sudden the notation is changed to $E_\lambda=E(\lambda)$.
Now, given an spectral measure $E$, one defines the spectral integral with respect to $v,w\in\mathcal{H}$ of the measurable function $f$ as the Lebesgue-Stieltjes integral $\int f(\lambda)d\langle E_\lambda v,w\rangle$, where $\langle\cdot,\cdot\rangle$ is the inner product in $\mathcal{H}$.
What is $\lambda$ supposed to represent? For me is weird to denote a Borel set with $\lambda$, but since that's the domain of $E$, I guess that's ok. But when I try to understand the integral, everything looks blurred. My only idea is that maybe $E_\lambda=E(\{\lambda\})$, is that so? Or am I missing something here?