TITLE: What's the relationship between a measure space and a metric space?
QUESTION [42 upvotes]: Definition of Measurable Space:
  An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$.
Definition of Measure:
  Let $(\Omega, F)$ be a measurable space, $μ$ is an non-negative function defined on $\mathcal{F}$ (that is $\mu: \mathcal{F} \to [0, +\infty]$). If $\mu(\emptyset) = 0$ and $\mu$ is countably additive (that is $A_n \in \mathcal{F}$, $n \geqslant 1$, $A_n \cap A_m = \emptyset$, $n \neq m \Rightarrow \mu(\cup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} \mu(A_n)$) then $\mu$ is a measure on $(\Omega, \mathcal{F})$.
Definition of Measure Space:
  Let μ is a measure on $(\Omega, \mathcal{F})$ then $(\Omega, \mathcal{F}, \mu)$ is a measure space.
Definition of Metric Space: A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e., a function $$d \colon M \times M \rightarrow \mathbb{R}$$ such that for any $x, y, z \in M$, the following holds: 

$d(x,y) \ge 0$    (non-negative),
$d(x,y) = 0\, \iff x = y\ $,     (identity of indiscernibles),
$d(x,y) = d(y,x)\ $,     (symmetry),
$d(x,z) \le d(x,y) + d(y,z)$     (triangle inequality).


Is a measure space $(\Omega, \mathcal{F}, \mu)$ necessarily a metric space?
What's the relationship between them?

REPLY [15 votes]: There is a theory of "metric measure spaces" which are metric spaces with a Borel measures, ie., a measure compatible with the topology of the metric space.  It has a big literature that is well represented online.
There is also the trivial answer, that if you don't demand compatibility of the measure and the metric, there is no particular relation between them.  A wider neighborhood has bigger measure, and easy things like that, but not necessarily anything nontrivial.