# Measure and metric

What, in simple terms, is the difference between a measure and a metric, and what applications require a measure rather than a metric, and vice versa? Examples would be great.

• Maybe looking at a book like Royden would immediately satisfy you? Dec 18 '18 at 18:37

Let $$X$$ be some set.

A metric is a function $$d: X \times X \rightarrow [0, \infty)$$ that describes the distance between the two input values. A function $$d$$ of this type is a metric, if it is positive definite, symmetric and fulfils the triangular inequality. Examples are the Euclidean distance on $$X = \mathbb{R}^n:$$ $$d(x,y)= \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$$. Or the discrete metric $$d(x,y)$$ that is $$0$$, if $$x=y$$ and $$1$$ otherwise.

A measure is defined on subsets of $$X$$, a so-called $$\sigma$$-algebra, say $$\mathcal{A}$$. This $$\sigma$$-algebra is a subset of the power set of $$X$$. While a metric measures distance, a measure typically measures area or volume of a set. In a 1D example, also sort of a distance. The Lebesgue measure in 2D for instance assigns each rectangle in $$\mathbb{R}^2$$ the area of the rectangle. Starting from this, one can show that it can be used to measure the area of most subsets of $$\mathbb{R}^2$$.

A metric measures the distance between two things. For instance, the usual metric in $$\mathbb{R}^2$$ is the for which the distance between $$(a,b)$$ and $$(c,d)$$ is $$\sqrt{(a-c)^2+(b-d)^2}$$.

A measure measures sizes of sets. For instance, with respect to the Lebesgue measure on $$\mathbb R$$, the measure of an interval $$[a,b]$$ is equal to $$b-a$$.