I was reading about the different metric vs. abstract open set definitions of topologies and wondered, whether it is possible to define a measure on ( R n , B n ), such that this measure is also a metric.Of course, in R the Lebesgue measure λ and the Euclidean distance d coincide, in the sense that for any interval [a,b), we have λ [ a , b ) = d ( a , b ). But with n ≥ 2 this is obviously not the case.I guess we would define a measure and then prove that it also satisfies the definition of a metric, but I don't know where to start in constructing such a measure.

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I was reading about the different metric vs. abstract open set definitions of topologies and wondered, whether it is possible to define a measure on (            R        n    ,            B        n    ), such that this measure is also a metric.Of course, in       R   the Lebesgue measure   λ and the Euclidean distance d coincide, in the sense that for any interval [a,b), we have   λ  [  a  ,  b  )  =  d  (  a  ,  b  ). But with   n  ≥  2 this is obviously not the case.I guess we would define a measure and then prove that it also satisfies the definition of a metric, but I don&#039;t know where to start in constructing such a measure.

Nirdaciw3 · Accepted Answer

A measure is not a distance and the converse is not true as well, simply by definition. The best example I can think of is the following where you use a measure on             R        n   to actually define a distance between sets.Denote with       |    ⋅      |   the   n−dimensional Lebesgue measure in             R        n  . Fix a set   A  ⊂            R        n  , a point   x  ∈  ∂  A,       H    x   the upper hyperplane passing through   x and a radius   r. Define then the following quantity:  D  (  x  ,  r  )  :=                    |            (      A      △              H        x            )      ∩              B                  r                    (      x      )              |                            α        n                    r        n            where       α    n   is the   n−dimensional measure of the unit ball.If you draw you see that the following quantity is exploiting a measure in             R        n   to define a distance (up to measure zero) between any set   A and the hyperplane passing through point   x (if we let them vary).

I was reading about the different metric vs. abstract open set definitions of topologies and wondere

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