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 $(\Omega, \mathcal{F})$ 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?