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Almost sure

by , PhD

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.

Table of Contents

Synonyms and acronyms

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


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.


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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How to cite

Please cite as:

Taboga, Marco (2021). "Almost sure", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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