Measure

A measure $\mu$ on a measurable space $(\mathrm{\Omega },\mathcal{F})$ is a function:

satisfying:

1. Null empty set:$$\mu (\mathrm{\varnothing })=0.$$
2. Countable additivity:
   For any pairwise disjoint sets $A_1,A_2,\dots \in \mathcal{F}$,$$\mu \left(\bigcup _{i=1}^{\mathrm{\infty }}{A}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mu (A_i).$$

A triple $(\mathrm{\Omega },\mathcal{F},\mu )$ is called a measure space.

Probability measure

A probability measure $P$ is a measure with two additional restrictions:

1. Normalization:$$P(\mathrm{\Omega })=1\text{(the total "mass" is 1)}.$$
2. Finite range:$$P(A)\in [0,1]\text{for all }A\in \mathcal{F}\text{(instead of }[0,\mathrm{\infty }]).$$

A triple $(\mathrm{\Omega },\mathcal{F},P)$ is called a probability space or probabilistic space.