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. The ordered pair is called a measurable space.

A σ-algebra of subsets is a set algebra of subsets; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[2]

The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. 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.

In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[3] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.

If one possible σ-algebra on is where is the empty set. In general, a finite algebra is always a σ-algebra.

If is a countable partition of then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.

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).

Motivation

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There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.

Measure

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A measure on   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. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets.

One would like to assign a size to every subset of   but in many natural settings, this is not possible. 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   These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.

Limits of sets

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Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.

  • The limit supremum or outer limit of a sequence   of subsets of   is   It consists of all points   that are in infinitely many of these sets (or equivalently, that are in cofinally many of them). That is,   if and only if there exists an infinite subsequence   (where  ) of sets that all contain   that is, such that  
  • The limit infimum or inner limit of a sequence   of subsets of   is   It consists of all points that are in all but finitely many of these sets (or equivalently, that are eventually in all of them). That is,   if and only if there exists an index   such that   all contain   that is, such that  

The inner limit is always a subset of the outer limit:   If these two sets are equal then their limit   exists and is equal to this common set:  

Sub σ-algebras

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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. Formally, if   are σ-algebras on  , then   is a sub σ-algebra of   if  . An example will illustrate how this idea arises.

Imagine two people are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads ( ) or Tails ( ). 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   or    

The observed information after   flips have occurred is one of the   possibilities describing the sequence of the first   flips. This is codified as the sub σ-algebra  

which locks down the first   flips and is agnostic about the result of the remaining ones. Observe that then   where   is the smallest σ-algebra containing all the others.

Definition and properties

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Definition

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Let   be some set, and let   represent its power set. Then a subset   is called a σ-algebra if and only if it satisfies the following three properties:[4]

  1.   is in  .
  2.   is closed under complementation: If some set   is in   then so is its complement,  
  3.   is closed under countable unions: If   are in   then so is  

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).

It also follows that the empty set   is in   since by (1)   is in   and (2) asserts that its complement, the empty set, is also in   Moreover, since   satisfies condition (3) as well, it follows that   is the smallest possible σ-algebra on   The largest possible σ-algebra on   is  

Elements of the σ-algebra are called measurable sets. An ordered pair   where   is a set and   is a σ-algebra over   is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to  

A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below).

Dynkin's π-λ theorem

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This theorem (or the related monotone class theorem) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

  • A π-system   is a collection of subsets of   that is closed under finitely many intersections, and
  • A Dynkin system (or λ-system)   is a collection of subsets of   that contains   and is closed under complement and under countable unions of disjoint subsets.

Dynkin's π-λ theorem says, if   is a π-system and   is a Dynkin system that contains   then the σ-algebra   generated by   is contained in   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   of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets in   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. For example, it is used to equate a probability for a random variable   with the Lebesgue-Stieltjes integral typically associated with computing the probability:   for all   in the Borel σ-algebra on   where   is the cumulative distribution function for   defined on   while   is a probability measure, defined on a σ-algebra   of subsets of some sample space  

Combining σ-algebras

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Suppose   is a collection of σ-algebras on a space  

Meet

The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by:  

Sketch of Proof: Let   denote the intersection. Since   is in every   is not empty. Closure under complement and countable unions for every   implies the same must be true for   Therefore,   is a σ-algebra.

Join

The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates a σ-algebra known as the join which typically is denoted   A π-system that generates the join is   Sketch of Proof: By the case   it is seen that each   so   This implies   by the definition of a σ-algebra generated by a collection of subsets. On the other hand,   which, by Dynkin's π-λ theorem, implies  

σ-algebras for subspaces

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Suppose   is a subset of   and let   be a measurable space.

  • The collection   is a σ-algebra of subsets of  
  • Suppose   is a measurable space. The collection   is a σ-algebra of subsets of  

Relation to σ-ring

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A σ-algebra   is just a σ-ring that contains the universal set  [5] A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite.

Typographic note

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σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus   may be denoted as   or  

Particular cases and examples

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Separable σ-algebras

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A separable  -algebra (or separable  -field) is a  -algebra   that is a separable space when considered as a metric space with metric   for   and a given finite measure   (and with   being the symmetric difference operator).[6] Any  -algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue  -algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).

A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. 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. If the measure space is separable, it can be shown that the corresponding metric space is, too.

Simple set-based examples

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Let   be any set.

  • The family consisting only of the empty set and the set   called the minimal or trivial σ-algebra over  
  • The power set of   called the discrete σ-algebra.
  • The collection   is a simple σ-algebra generated by the subset  
  • The collection of subsets of   which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of   if and only if   is uncountable). This is the σ-algebra generated by the singletons of   Note: "countable" includes finite or empty.
  • The collection of all unions of sets in a countable partition of   is a σ-algebra.

Stopping time sigma-algebras

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A stopping time   can define a  -algebra   the so-called stopping time sigma-algebra, which in a filtered probability space describes the information up to the random time   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   is  [7]

σ-algebras generated by families of sets

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σ-algebra generated by an arbitrary family

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Let   be an arbitrary family of subsets of   Then there exists a unique smallest σ-algebra which contains every set in   (even though   may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing   (See intersections of σ-algebras above.) This σ-algebra is denoted   and is called the σ-algebra generated by  

If   is empty, then   Otherwise   consists of all the subsets of   that can be made from elements of   by a countable number of complement, union and intersection operations.

For a simple example, consider the set   Then the σ-algebra generated by the single subset   is   By an abuse of notation, when a collection of subsets contains only one element,     may be written instead of   in the prior example   instead of   Indeed, using   to mean   is also quite common.

There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

σ-algebra generated by a function

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If   is a function from a set   to a set   and   is a  -algebra of subsets of   then the  -algebra generated by the function   denoted by   is the collection of all inverse images   of the sets   in   That is,  

A function   from a set   to a set   is measurable with respect to a σ-algebra   of subsets of   if and only if   is a subset of  

One common situation, and understood by default if   is not specified explicitly, is when   is a metric or topological space and   is the collection of Borel sets on  

If   is a function from   to   then   is generated by the family of subsets which are inverse images of intervals/rectangles in    

A useful property is the following. Assume   is a measurable map from   to   and   is a measurable map from   to   If there exists a measurable map   from   to   such that   for all   then   If   is finite or countably infinite or, more generally,   is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true.[8] Examples of standard Borel spaces include   with its Borel sets and   with the cylinder σ-algebra described below.

Borel and Lebesgue σ-algebras

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An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets.

On the Euclidean space   another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on   and is preferred in integration theory, as it gives a complete measure space.

Product σ-algebra

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Let   and   be two measurable spaces. The σ-algebra for the corresponding product space   is called the product σ-algebra and is defined by  

Observe that   is a π-system.

The Borel σ-algebra for   is generated by half-infinite rectangles and by finite rectangles. For example,  

For each of these two examples, the generating family is a π-system.

σ-algebra generated by cylinder sets

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Suppose  

is a set of real-valued functions. Let   denote the Borel subsets of   A cylinder subset of   is a finitely restricted set defined as  

Each   is a π-system that generates a σ-algebra   Then the family of subsets   is an algebra that generates the cylinder σ-algebra for   This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of   restricted to  

An important special case is when   is the set of natural numbers and   is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets   for which   is a non-decreasing sequence of σ-algebras.

Ball σ-algebra

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The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than the Borel σ-algebra. Note that the two σ-algebra are equal for separable spaces. For some nonseparable spaces, some maps are ball measurable even though they are not Borel measurable, making use of the ball σ-algebra useful in the analysis of such maps.[9]

σ-algebra generated by random variable or vector

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Suppose   is a probability space. If   is measurable with respect to the Borel σ-algebra on   then   is called a random variable ( ) or random vector ( ). The σ-algebra generated by   is  

σ-algebra generated by a stochastic process

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Suppose   is a probability space and   is the set of real-valued functions on   If   is measurable with respect to the cylinder σ-algebra   (see above) for   then   is called a stochastic process or random process. The σ-algebra generated by   is   the σ-algebra generated by the inverse images of cylinder sets.

See also

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References

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  1. ^ Elstrodt, J. (2018). Maß- Und Integrationstheorie. Springer Spektrum Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57939-8
  2. ^ "11. Measurable Spaces". Random: Probability, Mathematical Statistics, Stochastic Processes. University of Alabama in Huntsville, Department of Mathematical Sciences. Retrieved 30 March 2016. Clearly a σ-algebra of subsets is also an algebra of subsets, so the basic results for algebras in still hold.
  3. ^ Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1-118-12237-2.
  4. ^ Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.
  5. ^ Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 978-0-470-31795-2.
  6. ^ Džamonja, Mirna; Kunen, Kenneth (1995). "Properties of the class of measure separable compact spaces" (PDF). Fundamenta Mathematicae: 262. If   is a Borel measure on   the measure algebra of   is the Boolean algebra of all Borel sets modulo  -null sets. If   is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that   is separable if and only if this metric space is separable as a topological space.
  7. ^ Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras". Statistics and Probability Letters. 83 (1): 345–349. arXiv:1112.1603. doi:10.1016/j.spl.2012.09.024.
  8. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.
  9. ^ van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York. https://doi.org/10.1007/978-1-4757-2545-2
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