What is measure?

A measure on $(X, \mathcal A)$ is a map $\mu \colon \mathcal A \to [0, +\infty]$ with $\mu(\varnothing) = 0$ that is $\sigma$-additive: for pairwise disjoint $(A_n)_{n\in\N}$,

$(X, \mathcal A, \mu)$ is a measure space; $\mu$ is finite if $\mu(X) < \infty$, a probability measure if $\mu(X) = 1$, $\sigma$-finite if $X$ is a countable union of sets of finite measure. Examples: counting measure on $(\N, \mathcal P(\N))$; the Dirac mass $\delta_a(A) = \mathbf 1_{a \in A}$; and, the object of this chapter, Lebesgue measure.