# I was wondering: a measure &#x03BC;<!-- μ --> is a function that takes a set of numbers S

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 $\mu$.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Tucker House
One can certainly consider vector valued measures. Just take two measures ${\mu }_{1},{\mu }_{2}$ on the same measure space and set $\mu \left(E\right)=\left({\mu }_{1}\left(E\right),{\mu }_{2}\left(E\right)\right)\in {\mathbb{R}}_{+}^{2}$.