Spectral theory for sum of unbounded, commutative operators

Question

Let's suppose we have two unbounded operators $A$ and $B$ acting on Hilbert space $\mathcal{H}$. In order to simplify everything, let's suppose that $[A, B] = 0$. We have their spectral decomposition $A = \int a \, dP(a)$ and $B = \int b \, dP(b)$. If we look at the operator $A+B$, we have $A+B = \int (a+b) \, dP(a)dP(b)$. Is it possible to use the usual integration rules, so changing variables to $c = a+b$ and having the determinant of the jacobian matrix J, such that $C \equiv A + B =\int |J| \, c \, dP(c) $?