Measure

Statements

Definition. Let $\Omega$ be a set and $\mathcal{F}$ a σ-algebra over $\Omega$. A set function $P:\mathcal{F}\to \mathbb{R}\cup \{\infty \}$ is said to be a measure if it satisfies the following properties:

1. Non-negativity: For all $F\in \mathcal{F}$ , $P(F)\ge 0$
2. Null empty set: $P(\varnothing )=0$.
3. Countable additivity (or σ-additivity): For all countable collections $\{F_i{\}}_{i\in \mathbb{N}}$ of pairwise disjoint sets in $F$, the equality

$$P\left(\bigcup _{i\in \mathbb{N}}\right)=\sum _{i\in \mathbb{N}}^{\infty }P(F_i)$$

holds. The elements of $\mathcal{F}$ are said to be measurable sets.

References

Measure (mathematics) - Wikipedia