Editing Probability space (section)

=== Non-atomic examples ===

==== Example 4 ====
A number between 0 and 1 is chosen at random, uniformly. Here Ω = [0,1], <math> \mathcal{F}</math> is the σ-algebra of [[Borel set]]s on Ω, and ''P'' is the [[Lebesgue measure]] on [0,1].

In this case, the open intervals of the form {{open-open|''a'',''b''}}, where {{math|0 < ''a'' < ''b'' < 1}}, could be taken as the generator sets. Each such set can be ascribed the probability of {{math|1=''P''((''a'',''b'')) = (''b'' − ''a'')}}, which generates the [[Lebesgue measure]] on [0,1], and the [[Borel σ-algebra]] on Ω.

==== Example 5 ====
A fair coin is tossed endlessly. Here one can take Ω = {0,1}<sup>∞</sup>, the set of all infinite sequences of numbers 0 and 1. [[Cylinder set]]s {{math|1={(''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) ∈ Ω : ''x''<sub>1</sub> = ''a''<sub>1</sub>, ..., ''x''<sub>''n''</sub> = ''a''<sub>''n''</sub>}<nowiki/>}} may be used as the generator sets. Each such set describes an event in which the first ''n'' tosses have resulted in a fixed sequence {{math|(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}}, and the rest of the sequence may be arbitrary. Each such event can be naturally given the probability of 2<sup>−''n''</sup>.

These two non-atomic examples are closely related: a sequence {{math|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) ∈ {0,1}<sup>∞</sup>}} leads to the number {{math|2<sup>−1</sup>''x''<sub>1</sub> + 2<sup>−2</sup>''x''<sub>2</sub> + ⋯ ∈ [0,1]}}. This is not a [[one-to-one correspondence]] between {0,1}<sup>∞</sup> and [0,1] however: it is an [[standard probability space|isomorphism modulo zero]], which allows for treating the two probability spaces as two forms of the same probability space. In fact, all non-pathological non-atomic probability spaces are the same in this sense. They are so-called [[standard probability space]]s. Basic applications of probability spaces are insensitive to standardness. However, non-discrete conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure.