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Let $X=R_d \times R$, where $R_d$ denotes the set of real numbers with the discrete topology and $R$ the set of real numbers with the natural topology. For every $f \in C_c(X)$, one has $f(\{x\} \times R ) \neq 0$ for at most a finite number of $x_1, ...,x_m \in R$. We put $I(f)=\sum_{i=1}^m \int_{R} f(x_i, y)\text{d}y$. Then $I$ is a positive linear functional on $X$. Let $\mu$ be the measure corresponding to $I$ by the Riesz representation theorem.

How to show that $\mu(R_d \times \{0\})=\infty$ (or more generally, $\mu(A \times \{0\})=\infty$ if $A$ is not countable)?

Thanks.

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    Note: this gives the Haar measure for the topological group $\mathbb{R}_d \times \mathbb{R}$.2011-08-08

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How does the Riesz representation theorem go? Here is one way to do it:
First define $I$ for all nonnegative continuous functions $g$ like this: $ I(g) = \sup\{I(f):f \le g, f \in C_c(X)\} . $ Then define $\mu^*$ on open sets $G$ like this: $ \mu^*(G) = \inf\{I(g): g \ge 1_G, g \text{ continuous}\} . $ Then define $\mu^*$ for all sets $A$ like this: $ \mu^*(A) = \inf\{\mu^*(G): G\supseteq A, G\text{ open}\} . $ Finally restrict $\mu^*$ to its measurable sets.
So what do you get?

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    I used with:http://math.stackexchange.com/questions/56483/does-open-set-in-r-d-times-r-containing-r-times-0-contain-any-subset-i-ti/56486#564862011-08-11