# Is a metric/distance not a measure?

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."

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.

• 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? Commented Jul 13 at 9:32
• 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?. Commented Jul 13 at 11:19
• 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. Commented Jul 13 at 11:27
• @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.) Commented Jul 13 at 16:56
• 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. Commented Jul 13 at 18:11

• "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. Commented Jul 14 at 10:35