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A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might have a couple of further properties that must hold for the respective function to be a metric.

A measure (https://en.wikipedia.org/wiki/Measure_(mathematics)) on the other hand takes elements of a set's $\sigma$-algebra and maps those to a real number (or maybe even a complex number).

Let us assume that $X$ is the space over which I define my distance. Then, usually, the question "What is the distance between the three points $a, b, c \in X$?" does not make any sense, as a distance is only defined for a pair of elements of $X$.

The $\sigma$-algebra of $X$ must include the complement of each of the $\sigma$-algebra's elements; thus, the $\sigma$-algebra of $X$ must include elements (sets) that are not pairs, but include more than two elements of $X$.

As a result, it seems to me, that a distance is not a measure. This is bewildering to me, as I thought that "measure" is a generalization of "distance" (and other concepts).

What am I missing?

EDIT:

The misunderstanding might come from the number of times that "measure" (verb) is used in the Wikipedia text for "metric". E.g.: "The distance is measured by a function called a metric or distance function."

Maybe I should think about this like so:

If one says "the distance is measuring the length of the line", this actually be translated like to:

There is a line, which has the end points A and B. We want to measure the measure "length" of the line, which is a measure. While the line does have a length, it is difficult to measure it directly. However, we can simply calculate the distance between the points A and B (not the "distance of the line") and use that distance as a means to calculate the length of the line. The line has a length, its end-points A and B have a distance.

This "translation" is by me.

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    $\begingroup$ Going back to the definition, a distance in $X$ is a map $X\times X \to [0,+\infty)$. A measure is a map $\mathcal{A} \to [0,+\infty]$ where $\mathcal{A}$ is some $\sigma$-algebra on $X$. They are not even defined on the same space, how can one be a generalisation of the other? $\endgroup$
    – Didier
    Commented Jul 13 at 9:32
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    $\begingroup$ Somewhat relevant are notions of arc length measures for curves and surface area measures for curved surfaces and the like. More generally for $d<n$ a "$d$-dimensional measure" is applied to $d$-dimensional sets that can be "curved and twisted" though an $n$-dimensional ambient space. Although what follows is advanced, maybe you can get something from the google search Caratheodory + "linear measure" and the references in my answer to How is area defined?. $\endgroup$
    – Dave L. Renfro
    Commented Jul 13 at 11:19
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    $\begingroup$ Whoever said "measure is a generalization of distance" instead of "measure is a generalization of length/area" simply made a mistake/typo, and it seems your question is predicated on that one source's error. $\endgroup$
    – Mark S.
    Commented Jul 13 at 11:27
  • $\begingroup$ @MarkS.: So, "length" is does not refer to the notion of (i) "the length of a line, from point A to point B", more refers to the notion (ii) "the length of a thing that is made up from a series of points next to each other". Is that correct? (I purposefully do mean to use mathematical language in the formulations of the two notions, but I am trying to use everyday usage language.) $\endgroup$
    – Make42
    Commented Jul 13 at 16:56
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    $\begingroup$ This is a good example of an important linguistical/mathematical principle: there are not enough words in natural language to use them without ambiguity in mathematical discussions. Words get re-used in mathematics, sometimes over and over and over. So when you see a word used to formally express a mathematical concept, read the definition of that usage carefully. And when you see a word used to intuitively discuss a mathematical concept, be very very wary. $\endgroup$
    – Lee Mosher
    Commented Jul 13 at 18:11

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"Measure" is meant as a measure of "volume", or "space", how much "volume" a set takes up in an ambient space. Thus, nothing to do with distance.

Keep in mind that the names given to precise concepts in mathematics are usually chosen to remind you of some intuitive concept related to the precise definition, but this intuition can also mislead you if, for example, the intuitive meaning of the word is too vague or there are multiple meanings for that word in natural language. That's why we have precise definitions in the first place. Thus you shouldn't base yourself on the word itself to define the concept, the word should just be a mnemonic.

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    $\begingroup$ "Nothing to do with distance" may be a little strong: you might say that on $\mathbb R$ a measure of the interval $(a,b)$ (with $b>a$) is $b-a$, the distance between $a$ and $b$. In a sense this is the start of the ideas of measure and metrics which are then generalised in different ways to other less simple cases. $\endgroup$
    – Henry
    Commented Jul 14 at 10:35

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