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Almost sure

In probability theory, a property is said to hold almost surely if it holds for all sample points, except possibly for some sample points forming a subset of a zero-probability event.

If an event happens almost surely, then it is called an almost sure event.

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Synonyms and acronyms

The abbreviations a.s. (almost sure or almost surely) and w.p. 1 (with probability 1) are used very frequently.

Definition

The concept is defined as follows.

Definition Let Omega be a sample space. Let F be the set of its points satisfying a given property:[eq1]Property $Phi $ is said to be an almost sure property if the set $F^{c}$ of all points that do not satisfy property $Phi $ is included in a zero-probability event E, i.e.,[eq2]

Remember that not all subsets of the sample space are necessarily considered events (see the lecture entitled Probability). Therefore, the set F needs not necessarily be an event. In case it is, it is called an almost sure event.

Example

Suppose the sample space is the unit interval, that is,[eq3]and suppose that the set[eq4]is a zero-probability event:[eq5]Now consider the function[eq6]The function is well-defined on the interior of Omega but not at its endpoints (0 and 1), where it tends to infinity. Can we say that [eq7] is almost surely well-defined on Omega? Define the set[eq8]Then the complement of F is[eq9]Clearly, we have that[eq10]and we know that E is a zero-probability event. As a consequence, we can say that [eq11] is almost surely well-defined on Omega.

More details

A more articulated explanation, as well as an extensive discussion of the subtleties of the concept of zero-probability event, is provided in the lecture entitled Zero-probability events.

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