I have Borel measures $\mu$, $\nu$ on $\mathbb R^2$ with densities $g, h$ respectively with respect to the Lebesgue measure. Now I assume that $\mu(A \times \mathbb R) = \nu(A \times \mathbb R)$ for every Borel set $A$. Now I want to find a Borel measure $\gamma$ on $\mathbb R^3$ such that:
$$\mu(A \times B) = \gamma(A \times B \times \mathbb R)$$
$$\nu(A \times B) = \gamma(A \times \mathbb R \times B)$$
Fine. I try to write the measure $\gamma$ as
$$\gamma(Z) = \int 1_Z(x,y,z) \, d \nu(x,y) \, dz$$ and the same with $\mu$ instead of $\nu$, then I use the density wrt Lebesgue measure but this quickly becomes really messy. Is there an elegant way to do this?
Further, what would be the point of this exercise? Why would someone want this?
This is homework, so only hints.