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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.

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    By the way: this question reproduces http://math.stackexchange.com/q/5230/6179 (and note that Arturo's assertion in the comments on the main question is later on withdrawn in the comments on Byron's answer).2011-05-20

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There is no meaning to $\mu(x)$ (and Rudin does not use this notation). On the other hand the integral of a function $f$ with respect to a measure $\mu$ is denoted in a variety of ways, and amongst these, $ \int f \mathrm{d}\mu=\mu(f)=\int f(x) \mathrm{d}\mu(x)=\int f(x) \mu(\mathrm{d}x). $

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    Once again: you will not find $\mu(x)$ anywhere. What some people use is $\mathrm{d}\mu(x)$.2011-05-20