Given two spectral measures $E_{1},E_{2}$ defined in a given Hilbert space $\mathcal{H}$, define the following function $$ \mu(M \times N) = \langle E_{1}(A)E_{2}(B)x,x \rangle \quad \forall A,B \in \mathcal{B}_{\mathbb{R}}$$ where $x$ is fixed vector in $\mathcal{H}$. How do you prove $\mu$ is $\sigma$-additive?, it's similar to proving that a product measure is well-defined but I cannot find a way to adapt it to this case. The formal statement is
If $A \times B = \bigcup_{i=1}^{\infty}(A_{i} \times B_{i})$, $\quad A,B,A_{i}\text{'s},B_{i}\text{'s} \in \mathcal{B}_{\mathbb{R}}$, $\{ A_{i} \times B_{i} \}_{i=1}^{\infty}$ pairwise disjoint, prove that $$ \mu(A \times B) = \sum_{i=1}^{\infty}\mu(A_{i} \times B_{i})$$ Does anyone know how to tackle it? Any hint or idea would be helpful.