I was wondering: a measure μ is a function that takes a set of numbers S ∈ R n and assign a non-negative number to it. I'm summary: μ : S → R + Does any of you know if there are measures like this: μ : S → R + 2 The final aim is being able to distinguish between countable dense sets and non dense sets, so a subset S ∈ Q can have a measure greater than zero if it's dense, at least in one of the dimensions of μ.Many thanks in advance!

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I was wondering: a measure   μ is a function that takes a set of numbers   S  ∈            R        n   and assign a non-negative number to it. I&#039;m summary:   μ  :  S  →            R        +  Does any of you know if there are measures like this:  μ  :  S  →            R        +    2  The final aim is being able to distinguish between countable dense sets and non dense sets, so a subset   S  ∈      Q   can have a measure greater than zero if it&#039;s dense, at least in one of the dimensions of   μ.Many thanks in advance!

Tucker House · Accepted Answer

One can certainly consider vector valued measures. Just take two measures       μ    1    ,      μ    2   on the same measure space and set   μ  (  E  )  =  (      μ    1    (  E  )  ,      μ    2    (  E  )  )  ∈            R        +    2  .

I was wondering: a measure μ is a function that takes a set of numbers S

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