σ-algebra

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In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.

(For a list of mathematical logic notation used in this article see Notation in Probability and Statistics and/or List of Logic Symbols.)

The definition implies that it also includes the empty subset and that it is closed under countable intersections.

The pair is called a measurable space or Borel space.

A σ-algebra is a type of algebra of sets. An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]

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 that 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,[2] 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 - the σ-algebra produced by this process is known as the Borel algebra on the real line, and can also be conceived as the smallest (i.e. "coarsest") σ-algebra containing all the open sets, or equivalently containing all the closed sets. It is foundational to measure theory, and therefore modern probability theory, and a related construction known as the Borel hierarchy is of relevance to descriptive set theory.

MotivationEdit

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

MeasureEdit

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 setsEdit

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 of a sequence   each of which is a subset of   is
     
  • The limit infimum of a sequence   each of which is a subset of   is
     
  • If, in fact,
     
    then the   exists as that common set.

Sub σ-algebrasEdit

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 that 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. A simple example suffices to illustrate this idea.

Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads ( ) or Tails ( ). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the sample space   must consist of all possible infinite sequences of   or  :

 

However, after   flips of the coin, you may want to determine or revise your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2n possibilities for the first   flips. Formally, since you need to use subsets of   this is codified as the σ-algebra

 

Observe that then

 
where   is the smallest σ-algebra containing all the others.

Definition and propertiesEdit

DefinitionEdit

Throughout,   will be a set and   will denote its power set. A subset   is called a σ-algebra if it has the following three properties:[3]

  1.   is closed under complementation in  : If   is an element of   then so is its complement  
    • In this context,   is considered to be the universal set.
  2.   contains   as an element:  
    • Assuming that (1) holds, this condition is equivalent to   containing the empty set:  
  3.   is closed under countable unions: If   are elements of   then so is their union  
    • Assuming that (1) and (2) hold, it follows from De Morgan's laws that this condition is equivalent to   being closed under countable intersections: If   are elements of   then so is their intersection  

Equivalently, a σ-algebra is an algebra of sets that is closed under countable unions.

The empty set   belongs to   because by (1),   is in   and so (2) implies 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 (below).

Dynkin's π-λ theoremEdit

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-algebrasEdit

Suppose that   is a collection of σ-algebras on a space  

  • 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.

  • 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 subspacesEdit

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-ringEdit

A σ-algebra   is just a σ-ring that contains the universal set  [4] 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 noteEdit

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

Particular cases and examplesEdit

Separable σ-algebrasEdit

A separable σ-algebra (or separable σ-field) is a σ-algebra   that is a separable space when considered as a metric space with metric   for   and a given measure   (and with   being the symmetric difference operator).[5] Note that any σ-algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue σ-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. Note that 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 examplesEdit

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   that 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 σ-algebrasEdit

A stopping time   can define a  -algebra  , the so-called  -Algebra of τ-past, 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  .[6]

σ-algebras generated by families of setsEdit

σ-algebra generated by an arbitrary familyEdit

Let   be an arbitrary family of subsets of   Then there exists a unique smallest σ-algebra that 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  

Then   consists of all the subsets of   that can be made from elements of   by a countable number of complement, union and intersection operations. If   is empty, then   since an empty union and intersection produce the empty set and universal set, respectively.

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,   one may write   instead of   if it is clear that   is a subset 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 functionEdit

If   is a function from a set   to a set   and   is a σ-algebra of subsets of   then the σ-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 that 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.[7] Examples of standard Borel spaces include   with its Borel sets and   with the cylinder σ-algebra described below.

Borel and Lebesgue σ-algebrasEdit

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). Note that 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-algebraEdit

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 setsEdit

Suppose

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

For 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.

σ-algebra generated by random variable or vectorEdit

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 processEdit

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 alsoEdit

ReferencesEdit

  1. ^ "Probability, Mathematical Statistics, Stochastic Processes". Random. University of Alabama in Huntsville, Department of Mathematical Sciences. Retrieved 30 March 2016.
  2. ^ Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1-118-12237-2.
  3. ^ Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 978-0-07-054234-1.
  4. ^ Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 978-0-470-31795-2.
  5. ^ 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.
  6. ^ 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.
  7. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 978-0-387-95313-7.

External linksEdit