Consider two $\sigma$-finite measure spaces $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$.In standard measure theory books they define product measure as follows:
$(\mu\times \nu)(E)=\int_X\nu(E_x)d\mu =\int_Y \mu(E^y)d\nu$.
I am trying to find the motivation behind this definition.One may be that $E_x$ is the length of the small strip cut out from the set by $x$ and $d\mu(x)$ denotes a small differential length which partitions the set $X$.So,multiplying them and then integrating which is basically adding we get the area within that set intuitively.The assumption $\sigma$-finite is needed to ensure that area calculated by taking vertical partitions is the same as taking horizontal partitions of $E$.Is my intuition correct?
