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I was reading about mathematical structure and came across the distinction of metric and measure as follows:

A measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
A metric: there is a notion of distance between points.

Question: Isn't metric a super category of measure? So a measure is a form of a metric?

Andrés E. Caicedo
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GENIVI-LEARNER
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2 Answers2

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No, a measure is not a kind of metric.

In the contexts you quote, a measure tells you the size of a set. A metric tells you the distance between two points.

Ethan Bolker
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  • noted. but isnt measure quantify the extent of something, so in a way a distance is extent of closeness of one object to another? – GENIVI-LEARNER Jul 26 '20 at 21:12
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    Terminology in mathematics, such as a "measure", has precisely defined meanings, distinct from the meaning these words may have in everyday language. Of course, in the sense of everyday English, a metric does measure something, namely the distance of two points, but "a measure" in a mathematical context is a different thing. – Thorgott Jul 26 '20 at 21:20
  • @Thorgott noted. So essentially measure theory pertains to measure on sets, right? – GENIVI-LEARNER Jul 26 '20 at 22:23
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    Yes. More precisely, we have an ambient space we are working in (which is a set) and what we measure are certain subsets of that space. – Thorgott Jul 26 '20 at 22:26
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    @Thorgott makes so much sense! – GENIVI-LEARNER Jul 27 '20 at 12:46
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While Ethan Bolker's answer is correct of course, there are contexts in which it is fruitful to think of a measure as a form of metric. Indeed, (to keep formalities to a minimum), if $\mu$ is a probability measure on a measurable space $X$, then the $\mu$-measure of the symmetric difference of two measurable subsets is a pseudometric on the $\sigma$-algebra of $X$; factoring out sets of zero distance gives the measure algebra $\mathcal{M}(X,\mu)$ and the associated distance $d_\mu$ is complete. While $\mu$ is different from $d_\mu$, by virtue of the fact that many constructions in measure theory factor through the measure algebra (e.g. $L^p$ spaces), one could argue that these two objects serve similar functions. (see e.g. Equivalent definition of weakly mixing or https://www1.essex.ac.uk/maths/people/fremlin/mt.htm). (The proposed conflation is along the lines of Hirsch's conflation of maps and manifolds in differential topology.)

Added on 9/5/24: A paper along these lines is Kolmogorov's "Complete metric Boolean algebras".

Alp Uzman
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