Editing Event (probability theory) (section)

==Events in probability spaces==<!--Section linked from lead-->

Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard [[probability distributions]], such as the [[normal distribution]], the sample space is the set of real numbers or some subset of the [[real numbers]]. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers [[Pathological (mathematics)|'badly behaved']] sets, such as those that are [[Nonmeasurable set|nonmeasurable]]. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as [[Joint probability|joint]] and [[conditional probability|conditional probabilities]], to work, it is necessary to use a [[sigma-algebra|&sigma;-algebra]], that is, a family closed under complementation and countable unions of its members. The most natural choice of [[Sigma-algebra|&sigma;-algebra]] is the [[Borel measure|Borel measurable]] set derived from unions and intersections of intervals. However, the larger class of [[Lebesgue measure|Lebesgue measurable]] sets proves more useful in practice.

In the general [[Measure theory|measure-theoretic]] description of [[probability space]]s, an event may be defined as an element of a selected [[sigma-algebra|{{sigma}}-algebra]] of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the {{sigma}}-algebra is not an event, and does not have a probability.  With a reasonable specification of the probability space, however, all {{em|events of interest}} are elements of the {{sigma}}-algebra.