# Probability space

## Redirected from Probability measure

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In mathematics, a **probability space** is a set *Ω*, together with a σ-algebra (also known as a *sigma field*) *A* on *Ω* and a **probability measure** *P* on that σ-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 σ-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* ∈ *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.de:Wahrscheinlichkeitsraum eo:Probablo-spaco fr:Espace probabiliséhe:מרחב הסתברותno:Sannsynlighetsromru:Вероятностное пространство

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