In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe"[1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.
This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities.
Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).
is a function that assigns a non-negative real number to subsets of
this can be thought of as making precise a notion of "size" or "volume" for sets.
For example, the axiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the Vitali sets.
For this reason, one considers instead a smaller collection of privileged subsets of
Non-empty collections of sets with these properties are called σ-algebras.
Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets.
In much of probability, especially when conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed.
This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information.
Imagine two people are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads (
Both players are assumed to be infinitely wealthy, so there is no limit to how long the game can last.
This means the sample space Ω must consist of all possible infinite sequences of
is called a σ-algebra if and only if it satisfies the following three properties:[4] From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).
Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in
enjoy the property under consideration while, on the other hand, showing that the collection
enjoy the property, avoiding the task of checking it for an arbitrary set in
One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals.
σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface.
-algebra generated by a countable collection of sets is separable, but the converse need not hold.
The symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric.
However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the induced metric.
in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often repeating it until the time
Then there exists a unique smallest σ-algebra which contains every set in
by a countable number of complement, union and intersection operations.
By an abuse of notation, when a collection of subsets contains only one element,
and is preferred in integration theory, as it gives a complete measure space.
the σ-algebra generated by the inverse images of cylinder sets.