question on a notation in Big Rudin

I am asking yourself why Rudin made use of the adhering to symbols in his "Real and also Complex Analysis". It remains in Definition 8.7, as adhering to.

If $(X, \mathscr{S}, \mu)$ and also $(Y, \mathscr{T}, \lambda)$ are 2 $\sigma$ - limited action rooms, and also $Q\in \mathscr{S}\times \mathscr{T}$, after that specify $(\mu\times \lambda)(Q)=\int_{X}{\lambda(Q_x) d\mu(x)}=\int_{Y}{\mu(Q^y) d\lambda(y)}$.

My inquiry is why $\mu$ and also $\lambda$ rely on $x$ and also $y$ specifically? Aren't they dealt with as soon as the action rooms $(X, \mathscr{S}, \mu)$ and also $(Y, \mathscr{T}, \lambda)$ are offered? What is the definition of $\mu(x)$ and also $\lambda(y)$ or the writer intends to stress below?

Many thanks.

2022-06-07 14:41:36
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Answers: 1

There is no definition to $\mu(x)$ (and also Rudin does not utilize this symbols). On the various other hand the indispensable of a function $f$ relative to an action $\mu$ is represented in a selection of means, and also among these, $$ \int f \mathrm{d}\mu=\mu(f)=\int f(x) \mathrm{d}\mu(x)=\int f(x) \mu(\mathrm{d}x). $$

2022-06-07 15:10:37