TITLE: Measure and metric
QUESTION [12 upvotes]: 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.

REPLY [9 votes]: 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 $\mathbb{R}^2$.