Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the following representation holds $$ \mu_1\otimes\mu_2(A)=\int_{X_2}\mu_1(A^{x_2})d\mu_2(x_2)=\int_{X_1}\mu_2(A_{x_1})d\mu_1(x_1) $$ where $A_{x_1}$ and $A^{x_2}$ are the usual slices of $A$ in the two "directions" for fixed points $x_1,x_2$.

Now suppose to have two commuting projection valued measures $E_{1/2}:\mathcal{B}(X_{1/2})\rightarrow P(H)$ where $H$ is a complex separable Hilbert space, then there exists a unique joint projection valued measure $E:\mathcal{B}(X_1\times X_2)\rightarrow P(H)$ such that $$ E(A_1\times A_2)=E_1(A_1)E_2(A_2). $$

My question is: does there exist some "integral representation" of $E$ in terms of $E_1$ and $E_2$ similarly to the $\mu_1\otimes\mu_2$ case?

I would like to have some relation like $$ E(A)=\int_{X_2} E_1(A^{x_2})d E_2(x_2)=\int_{X_1} E_2(A_{x_1})d E_1(x_1) $$ although I don't know how to define this rigorously.