Probability Set Function

Definition

Let

• $\mathcal{C}$ : Sample space
• $\mathcal{B}$ : Events ($\mathcal{B}\subset \mathcal{C}$)
• $P:\mathcal{B}\to \mathbb{R}$ (Function)

If

1. $P(A)\ge 0,\forall A\subset \mathcal{B}$
2. $P(\mathcal{C})=1$
3. If $\{A_n\}\in \mathcal{B}$ and $A_m\cap A_n=\varphi ,\forall m\ne n$ then

$$P\left(\bigcup _{n=1}^{\infty }{A}_n\right)=\sum _{n=1}^{\infty }P\left(A_n\right)$$

Then

• We say $P$ is a probability set function
• We call the return value of $P$ as the probability

Remark

A probability set function is essentially a function that maps subset of events $\mathcal{B}$ to a real number.

It also must satisfy certain axioms (referred to as Kolmogorov axiom of probability). The conditions listed above can be interpreted as:

1. The probability of an event is a non-negative real number:
2. The probability of the sample space itself is 1
3. If events are mutually exclusive (i.e., cannot occur simultaneously), then the probability that at least one of them occurs equals the sum of their individual probabilities.