Let \(X\) be a set with a \(\sigma\)-algebra \(\mathcal{A}.\) A measure, \(\mu\), is a function \[\mu: \mathcal{A} \rightarrow [0,\infty)\] that satisfies the following:

1. \(\mu(\emptyset) = 0\)
2. If \(A_1, A_2, \dots\) are pairwise disjoint, \[\mu\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \mu(A_i)\]

Example

Let \(X = \mathbb{Z}\) with \(\sigma\)-algebra \(\mathcal{P}(\mathbb{Z}).\) The function \(\mu(A) = |A|\) is a measure on \(\mathbb{Z}\) called the counting measure.

To simplify, take for example \(A = \{-3,2,7\}.\) Then \(\mu(A) = 3\) since \(A\) has \(3\) elements.

Checking that \(\mu\) is a measure is almost trivial. First, for any set \(A \subset \mathbb{Z},\) \(\mu(A) \geq 0\) since sets contain \(0\) or more elements. Second, \(\mu(\emptyset ) = 0\) since \(\emptyset \) has no elements. Third, cardinality is additive. In particular, if \(A_1, A_2, \dots \subset \mathbb{Z}\) then \[\left| \bigcup_{i=1}^\infty A_i \right| = \sum_{i=1}^\infty |A_i|\]