Probability Measure

probability-theory measure-theory

Definition

Probability Measure

A probability measure $P:\mathfrak{S}\to [0,1]$ is a measure that assigns every possible event $A\in \mathfrak{S}$ ($\mathfrak{S}$ being a sigma algebra) a probability.

Relation to Probability Mass Function

From a probability mass function, a probability measures over a discrete space is given by summing over the outcomes:

$$P(A)=\sum _{x\in A}p(x)A\subseteq \mathfrak{S}$$