Measure

Concept

A measure is a generalization of concepts of length, area, volume and etc.

Definition

Let \(\Sigma\) denotes σ-algebra on the set \(X\).

\(\mu: \Sigma \mapsto \overline \Re\) is a measure on \(X\), where \(\overline \Re = \Re \cup \{-\infty, +\infty\}\).

Additionally, \(\mu\) should satisfies the following properties:

1. \[\forall E \in \Sigma: \mu(E) \geq 0\]
2. \[\mu(\emptyset) = 0\]
3. \[\mu(\bigcup_i E_i) = \sum_i \mu(E_i)\]