I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the Borel measurable sets on a Hilbert Space with some caveats ($P(0)=0$, $P(\mathbb{R})=1$, etc.) I also understand that for a self-adjoint operator, we have $P(\Lambda)=\chi_{\Lambda}(H)$ where $\chi$ is the characteristic function on the Borel set $\Lambda$.
In one of my questions, I was given a matrix that was just a linear combination of Pauli matrices and asked to find the projection valued measure of it. There was no mention of Borel sets or anything, so I became a little confused.
I found that for Hermitian matrices, we can use functional calculus to find $$f(H)=\sum\limits_{i} f(E_i)P_i$$
Where $f$ is a Borel function, $E_i$ the eigenvalues of $H$ and $P_i$ the corresponding projections into the eigenspaces. Is this the same as the Projection Valued Measure for a $n\times n$ Hermitian matrix?
EDIT: I think I might have figured it out. Wouldn't the projection valued measure just be $$P(\Lambda)=\sum\limits_{\{i|E_i\in\Lambda\}} P_i$$ Where $E_i$ are the eigenvalues and $P_i$ the projections into the eigenspaces? Again, no Borel sets are defined, but we are working in $\mathbb{C}^2$.
Typically, one has a locally compact or compact topological Hausdorff space $\Omega$, and the projection measure is defined on the Borel subsets of $\Omega$. In the context of the Spectral Theorem, $\Omega$ is the closed subset of $\mathbb{C}$ which is the spectrum of a normal operator $N$, but the abstraction to a more general space can be useful, especially when dealing with operator algebras.
In your case, the spectrum is discrete, and you'll use the subspace topology inherited from $\mathbb{C}$, which is the discrete topology. This is a compact topology. The projection measure is as you stated for this case: $E\{ \lambda\}$ is the orthogonal projection onto the eigenspace with eigenvalue $\lambda$ for the hermitian matrix $H$. Then $\int \lambda dE = H$.