A measure $\mu$ on a set $\Omega$, with Sigma-algebra $\mathcal{A}$, is a Function $\mu: \mathcal{A} \rightarrow [0, \infty]$, s.t.

1. $\mu(\emptyset)=0$
2. $\mu(A) \leq \mu(B)$ if $A,B \in \mathcal{A}$ and $A \subseteq B$
3. countable additivity. $\mu(\bigcup\limits_{i=1}^{\infty} E_i) = \sum\limits_{i=1}^{\infty} \mu(E_i)$ for any collection $E_1, E_2, ... \in \mathcal{A}$ of mutually disjoint sets.

Specifying a measure on a sigma-algebra is simplified by the

Types of measures

• Probability measures
• Lebesgue measure

Outer measure

Definition

(PP 1.4) Measure theory: Examples of Measures

(PP 1.5) Measure theory: Basic Properties of Measures