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I was wondering: a measure $\mu$ is a function that takes a set of numbers $S \in \mathbb{R}^n$ and assign a non-negative number to it. I'm summary: $\mu: S \rightarrow \mathbb{R}_+$

Does any of you know if there are measures like this:

$$\mu: S \rightarrow \mathbb{R}^2_+$$

The final aim is being able to distinguish between countable dense sets and non dense sets, so a subset $S \in \mathbb{Q}$ can have a measure greater than zero if it's dense, at least in one of the dimensions of $\mu$.

Many thanks in advance!

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  • $\begingroup$ You mean $S \subseteq \mathbb Q$ instead of $S \in \mathbb Q$. $\endgroup$
    – md2perpe
    Commented Jul 6, 2017 at 10:51

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One can certainly consider vector valued measures. Just take two measures $\mu_1, \mu_2$ on the same measure space and set $\mu (E) = (\mu_1 (E), \mu_2 (E) ) \in \mathbb R_+^2$.

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  • $\begingroup$ Yes, but beyond that fact, is there a measure that allows to differenciate between dense countable sets and non-dense sets? Because Lebesgue measure cannot. It assigns just 0 to both sets. $\endgroup$ Commented Jul 5, 2017 at 19:19
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    $\begingroup$ There is no measure $\mu $ being zero for finite sets (those are not dense) and non-zero for $\mathbb Q $ because of countable additivity. For the same reason I can not see any possibility to have a measure that differs between dense and non-dense sets. $\endgroup$
    – md2perpe
    Commented Jul 5, 2017 at 19:50

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