Measure and metric
Sophia Terry 
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.
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$\begingroup$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 $b-a$.
$\endgroup$ $\begingroup$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 $\mathbb{R}^2$.
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