My exact question is asked and answered here:
Proving that a spectral measure is $\sigma$-additive
but I have spent all day trying to construct the $A^{'}_js$ and $B^{'}_k$s described in the answer and haven't succeeded. I know A und B must each be elements of the Borel-sigma Algebra on $\mathbb{R} \ (\mathcal{B}(\mathbb{R})$) because the Borel-sigma algebra on $\mathbb{R^2}$ is the cartesian product of $\mathcal{B}(\mathbb{R})$ with itself, $$\mathcal{B}(\mathbb{R^2}) = \mathcal{B}(\mathbb{R}) \times \mathcal{B}(\mathbb{R})$$ and because of sigma-algebra qualities, the union over $A_i \times B_i$ must be contained in $\mathcal{B}(\mathbb{R^2})$, but I don't know exactly what A and B look like, and maybe that's my problem. I would really appreciate some help constructing the $A^{'}_js$ and $B^{'}_k$s.
