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.
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 be a sample space. Let be the set of its points satisfying a given property:Property is said to be an almost sure property if the set of all points that do not satisfy property is included in a zero-probability event , i.e.,
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.
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 .
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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