Probability space

In mathematics, a probability space is a set Ω, together with a &sigma;-algebra (also known as a sigma field) A on Ω and a probability measure P on that &sigma;-algebra; that is, a positive measure P on the measurable space (Ω, A) such that P(Ω) = 1. The measurable subsets of Ω, i.e., the sets belonging to the &sigma;-algebra A, are called events. The measure P is called the probability measure, and P(E) is the probability of the event E. If ω &isin; Ω, ω is an outcome.

If a measurable function X (or Y or Z or such -- capital Roman letters at the end of the alphabet are typically employed in this context) maps Ω into a space S from which we collect samples, the set S   is called the sample space. The measurable function X is termed a random variable.

To explain the difference between Ω and S, consider the following: Ω could be virtually anything. As examples, Ω could be {the people of earth}, {the atoms in the cosmos}, {all 18K gold jewelry or golden crowns}, or any such set of interest. But the sample space S would be something such as, respectively, the set of vectors (height, weight, age) of the people; the set of number of electrons of the atoms and their quantum energy levels; the set of all weights of the golden items. Note that not all subsets of a probability space are necessarily events.

Wahrscheinlichkeitsraum Probablo-spaco Espace probabilisé Misura di probabilità מרחב הסתברות 確率空間 Sannsynlighetsrom Przestrzeń probabilistyczna Вероятностное пространство