I am wondering why Rudin used the following notation in his "Real and Complex Analysis". It is in Definition 8.7, as following.
If $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ are two $\sigma$-finite measure spaces, and $Q\in \mathscr{S}\times \mathscr{T}$, then define $(\mu\times \lambda)(Q)=\int_{X}{\lambda(Q_x) d\mu(x)}=\int_{Y}{\mu(Q^y) d\lambda(y)}$.
My question is why $\mu$ and $\lambda$ depend on $x$ and $y$ respectively? Aren't they fixed once the measure spaces $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ are given? What is the meaning of $\mu(x)$ and $\lambda(y)$ or the author wants to emphasize here?
Thanks.