5. Measures

Definition 5.1 (Measure): Let $(\Omega, \sigmaF)$ be a measurable space. A function $\mu : \sigmaF \to \overline{\R}_{+}$ is said to be a measure on $\sigmaF$ if the following two items are satisfied:

1. $\mu(\varnothing) = 0$
2. Given any disjoint family $\qty{A_i : i \in \N} \subset \sigmaF$, we have that $ \mu\qty(\bigcup_{i\in \N} A_i) = \sum_{i \in \N} \mu(A_i)$.