How to interpret $\int_{\mathbb R}\lambda d\left<E_\lambda f,g\right>$ in spectral theorem?

Question

In spectral theorem, it's written : Let $-A:H\to H$ self adjoint with domain $D(A)$ and $H$ a Hilbert space. There is a spectral measure $(E_\lambda )_{\lambda \in\mathbb R}$ s.t. $$\left<-Af,g\right>=\int_{\mathbb R}\lambda d\left<E_\lambda f,g\right>,\quad f\in D(A),g\in H.$$

I really don't see how to interpret $\int_{\mathbb R}\lambda d\left<E_\lambda f,g\right>$ neither what is exactely a spectral measure. Someone has an idea ?

Answer 1

For fixed $f \in D(A)$ and $g \in H$ the function $\lambda \to \langle E_{\lambda} f, g \rangle$ corresponds to a measure $\mu_{f,g}$ via the formula $\mu_{fg} (-\infty, \lambda])=\langle E_{\lambda} f, g \rangle$. You are integrating w.r.t this measure.