What's the relationship between a measure space and a metric space?
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?
The best I can think of, are: Given a metric space $(X,d)$ , we can assign sigma-algebras.
1) Borel Measure: This is the sigma algebra generated by the open sets generated by the open balls in the metric.
Or, 2)Looking at some of the linked questions are that of a Hausdorff measure associated with a metric space:
https://en.wikipedia.org/wiki/Hausdorff_measure
But yours is an interesting question: given a measure triple (X, A, $\mu$), where $A$ is a sigma algebra and $X$ is the underlying space, can this be the Borel algebra resulting from a metric space? I don't have a full answer but some obvious requirements are that $X$ must be metric , or at least metrizable. Still, while outside of the scope of your question, one can define measures on non-metric, non-metrizable spaces using the Borel sigma algebra.
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.