[Math] What’s the relationship between a measure space and a metric space

analysislebesgue-measuremeasure-theoryreal-analysis

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:

  1. $d(x,y) \ge 0$ (non-negative),

  2. $d(x,y) = 0\, \iff x = y\ $, (identity of indiscernibles),

  3. $d(x,y) = d(y,x)\ $, (symmetry),

  4. $d(x,z) \le d(x,y) + d(y,z)$ (triangle inequality).

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

Best Answer

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.

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