Almost sure

by Marco Taboga, 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.

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.

Remember that not all subsets of the sample space are necessarily considered events (see the lecture entitled Probability). Therefore, the set 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,and suppose that the setis a zero-probability event:Now consider the functionThe function is well-defined on the interior of but not at its endpoints ( and ), where it tends to infinity. Can we say that is almost surely well-defined on ? Define the setThen the complement of isClearly, we have thatand we know that is a zero-probability event. As a consequence, we can say that is almost surely well-defined on .

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. https://www.statlect.com/glossary/almost-sure.