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: $\mu :S\to {\mathbb{R}}_{+}$Does any of you know if there are measures like this:

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

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!

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

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!