tõenäosusruumil

A probability space, also known as a probability triple, is a mathematical structure used to model random experiments. It consists of three components: a sample space, an event space, and a probability measure.

The sample space, denoted by Ω, is the set of all possible outcomes of a random experiment. Each element in Ω represents a possible outcome.

The event space, denoted by F, is a sigma-algebra of subsets of Ω. It represents the collection of all possible events, where an event is a subset of Ω. The sigma-algebra must satisfy certain properties, including being closed under countable unions, intersections, and complements.

The probability measure, denoted by P, is a function that assigns a probability to each event in F. It satisfies the following axioms: (1) P(E) ≥ 0 for all E in F, (2) P(Ω) = 1, and (3) if E1, E2, ... are pairwise disjoint events in F, then P(∪Ei) = ∑P(Ei).

Probability spaces are fundamental in probability theory and are used to define and analyze random variables, expectations, and other probabilistic concepts. They provide a rigorous framework for modeling uncertainty and randomness in various fields, including statistics, physics, and engineering.