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The title says it all, but just to be more specific:

Given a set $S$ with metric $d$, the measure $m$ would be defined in terms of $d$ and would have properties one would want a measure to have, such as monotonicity, subadditivity, and continuity.

while1fork
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    https://math.stackexchange.com/questions/1402847/whats-the-relationship-between-a-measure-space-and-a-metric-space – Elle Najt Sep 17 '20 at 19:19
  • @while1fork: Measure are metrizability may not be linked except for the domain of the measure. Typically, on metric spaces, a measure will be defined on Borel sets (the $\sigma$-algebra generated by open sets. But then again The Borel sets are not entirely dependent on a particular metric: given a metric, there are infinitely many metrics $d'$ that product the same topology, and here the same Borel $\sigma$-algebra (CONTINUE BELOW) – Mittens Sep 17 '20 at 22:01
  • @while1fork: Consider the counting measure on $(\mathbb{N},\mathscr{B}(\mathbb{N})$. IT has nothing to do with the distance $d(x,y)=|x-y|$. – Mittens Sep 17 '20 at 22:05

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