Probability Spaces and Events - Core Definitions

The foundation of modern probability theory rests on the measure-theoretic framework introduced by Kolmogorov in 1933. This axiomatic approach provides a rigorous mathematical foundation for understanding randomness and uncertainty.

Probability Space

Definition

A probability space is a triple $(\Omega, \mathcal{F}, P)$ where:

$\Omega$ is a non-empty set called the sample space representing all possible outcomes
$\mathcal{F}$ is a $\sigma$ -algebra on $\Omega$ , a collection of subsets called events
$P: \mathcal{F} \to [0,1]$ $P : F \to [0, 1]$ is a probability measure satisfying:
1. $P(\Omega) = 1$ (normalization)
2. For any countable collection of disjoint events $\{A_i\}_{i=1}^{\infty} \subseteq \mathcal{F}$ : $P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$ (countable additivity)

The $\sigma$ -algebra $\mathcal{F}$ must satisfy three properties: it contains $\Omega$ , is closed under complements, and is closed under countable unions. This structure ensures that we can perform all necessary set operations within the class of measurable events.

Events and Operations

Events are subsets of the sample space that belong to the $\sigma$ -algebra. For events $A, B \in \mathcal{F}$ :

The union $A \cup B$ represents " $A$ or $B$ occurs"
The intersection $A \cap B$ represents " $A$ and $B$ both occur"
The complement $A^c = \Omega \setminus A$ represents " $A$ does not occur"
Events $A$ and $B$ are disjoint (or mutually exclusive) if $A \cap B = \emptyset$

Example

Consider tossing a fair coin twice. The sample space is $\Omega = \{HH, HT, TH, TT\}$ . The $\sigma$ -algebra $\mathcal{F}$ contains all $2^4 = 16$ subsets. The event "at least one head" is $A = \{HH, HT, TH\}$ with probability $P(A) = 3/4$ .

Conditional Probability

Definition

Given events $A, B \in \mathcal{F}$ with $P(B) > 0$ , the conditional probability of $A$ given $B$ is: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

This fundamental concept allows us to update probabilities based on new information. It forms the basis for Bayes' theorem and is essential in statistical inference.

Remark

The axioms of probability were chosen to match our intuitive understanding while providing mathematical rigor. The requirement of countable (not just finite) additivity is crucial for working with continuous distributions and taking limits in probability theory.