Sigma-algebra

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions (the definition implies that it also includes the empty subset and that it is closed under countable intersections). The pair (X, Σ) 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 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,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.

If X = {a, b, c, d}, one possible σ-algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. In general, a finite algebra is always a σ-algebra.

If {A1, A2, A3, …} is a countable partition of X 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.

Motivation

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

Measure

A measure on X is a function that assigns a non-negative real number to subsets of X; 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 X, 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 X. 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

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 A1, A2, A3, ..., each of which is a subset of X, is
  • The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is
  • If, in fact,
then the exists as that common set.

Sub σ-algebras

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. 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 (H) or Tails (T). 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 Ω must consist of all possible infinite sequences of H or T:

However, after n 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 n flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra

Observe that then

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

Definition and properties

Definition

Let X be some set, and let represent its power set. Then a subset is called a σ-algebra if it satisfies the following three properties:[3]

  1. X is in Σ, and X is considered to be the universal set in the following context.
  2. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
  3. Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1A2A3 ∪ … .

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 Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set, is also in Σ. Moreover, since {X, ∅} satisfies condition (3) as well, it follows that {X, ∅} is the smallest possible σ-algebra on X. The largest possible σ-algebra on X is 2X:=.

Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebra over X, 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 [0, ∞].

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 theorem

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

Dynkin's π-λ theorem says, if P is a π-system and D is a Dynkin system that contains P then the σ-algebra σ(P) generated by P is contained in D. Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D 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 σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P).

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 X with the Lebesgue-Stieltjes integral typically associated with computing the probability:

for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while is a probability measure, defined on a σ-algebra Σ of subsets of some sample space Ω.

Combining σ-algebras

Suppose is a collection of σ-algebras on a space X.

  • 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 Σ denote the intersection. Since X is in every Σα, Σ is not empty. Closure under complement and countable unions for every Σα implies the same must be true for Σ. Therefore, Σ 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 n = 1, 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

Suppose Y is a subset of X and let (X, Σ) be a measurable space.

  • The collection {YB: B ∈ Σ} is a σ-algebra of subsets of Y.
  • Suppose (Y, Λ) is a measurable space. The collection {AX : AY ∈ Λ} is a σ-algebra of subsets of X.

Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[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 note

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

Particular cases and examples

Separable σ-algebras

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 examples

Let X be any set.

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

Stopping time sigma-algebras

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 sets

σ-algebra generated by an arbitrary family

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

If F is empty, then σ(F)={X, ∅}. Otherwise σ(F) consists of all the subsets of X that can be made from elements of F by a countable number of complement, union and intersection operations.

For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the single subset {1} is σ({{1}}) = {∅, {1}, {2, 3}, {1, 2, 3}}. By an abuse of notation, when a collection of subsets contains only one element, A, one may write σ(A) instead of σ({A}); in the prior example σ({1}) instead of σ({{1}}). Indeed, using σ(A1, A2, ...) to mean σ({A1, A2, ...}) 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

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 . i.e.

A function f from a set X to a set Y is measurable with respect to a σ-algebra Σ of subsets of X if and only if σ(f) is a subset of Σ.

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

If f is a function from X to Rn then σ(f) is generated by the family of subsets which are inverse images of intervals/rectangles in Rn:

A useful property is the following. Assume f is a measurable map from (X, ΣX) to (S, ΣS) and g is a measurable map from (X, ΣX) to (T, ΣT). If there exists a measurable map h from (T, ΣT) to (S, ΣS) such that f(x) = h(g(x)) for all x, then σ(f) ⊂ σ(g). If S is finite or countably infinite or, more generally, (S, ΣS) is a standard Borel space (e.g., a separable complete metric space with its associated Borel sets), then the converse is also true.[7] Examples of standard Borel spaces include Rn with its Borel sets and R with the cylinder σ-algebra described below.

Borel and Lebesgue σ-algebras

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

Product σ-algebra

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 Rn 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

Suppose

is a set of real-valued functions. Let denote the Borel subsets of R. A cylinder subset of X 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 X. This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of restricted to X.

An important special case is when is the set of natural numbers and X 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 vector

Suppose is a probability space. If is measurable with respect to the Borel σ-algebra on Rn then Y is called a random variable (n = 1) or random vector (n > 1). The σ-algebra generated by Y is

σ-algebra generated by a stochastic process

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 X, then Y is called a stochastic process or random process. The σ-algebra generated by Y is

the σ-algebra generated by the inverse images of cylinder sets.

See also

References

  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 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 iff 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. doi:10.1016/j.spl.2012.09.024.
  7. ^ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

External links

Ba space

In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is (Dunford & Schwartz 1958, IV.2.15)

If Σ is a sigma-algebra, then the space is defined as the subset of consisting of countably additive measures. (Dunford & Schwartz 1958, IV.2.16) The notation ba is a mnemonic for bounded additive and ca is short for countably additive.

If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then is the subspace of consisting of all regular Borel measures on X. (Dunford & Schwartz 1958, IV.2.17)

Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets).

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

Counting measure

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and ∞ if the subset is infinite.

The counting measure can be defined on any measurable set, but is mostly used on countable sets.

In formal notation, we can make any set X into a measurable space by taking the sigma-algebra of measurable subsets to consist of all subsets of . Then the counting measure on this measurable space is the positive measure defined by

for all , where denotes the cardinality of the set .

The counting measure on is σ-finite if and only if the space is countable.

Cylinder set

In mathematics, a cylinder set is the natural set in a product space. Such sets are the basis for the open sets of the product topology and, they are a generating family of the cylinder sigma-algebra, which in the countable case is the product sigma-algebra.

Cylinder sets are particularly useful in providing the base of the natural topology of the product of a countable number of copies of a set. If V is a finite set, then each element of V can be represented by a letter, and the countable product can be represented by the collection of strings of letters.

Cylindrical σ-algebra

In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra is a σ-algebra often used in the study either product measure or probability measure of random variables on Banach spaces.

For a product space, the cylinder σ-algebra is the one which is generated by cylinder sets. As for products of countable length, the cylindrical σ-algebra is the product σ-algebra.In the context of Banach space X, the cylindrical σ-algebra Cyl(X) is defined to be the coarsest σ-algebra (i.e. the one with the fewest measurable sets) such that every continuous linear function on X is a measurable function. In general, Cyl(X) is not the same as the Borel σ-algebra on X, which is the coarsest σ-algebra that contains all open subsets of X.

Doob–Dynkin lemma

In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the σ {\displaystyle \sigma } -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.

The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the σ {\displaystyle \sigma } -algebra that is generated by the random variable.

Filtration (mathematics)

In mathematics, a filtration is an indexed set of subobjects of a given algebraic structure , with the index running over some index set that is a totally ordered set, subject to the condition that

if in , then .

If the index is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic object gaining in complexity with time. Hence, a process that is adapted to a filtration , is also called non-anticipating, i.e. one that cannot see into the future.

Sometimes, as in a filtered algebra, there is instead the requirement that the be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy , where the index set is the natural numbers; this is by analogy with a graded algebra.

Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the be the whole , or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the to is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. This article does not impose this requirement.

There is also the notion of a descending filtration, which is required to satisfy in lieu of (and, occasionally, instead of ). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with cofiltrations (which consist of quotient objects rather than subobjects).

The concept dual to a filtration is called a cofiltration.

Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.

Generated σ-algebra

The generated σ-algebra or generated σ-field refers to

The smallest σ-algebra that contains a given family of sets, see Generated σ-algebra (by sets)

The smallest σ-algebra that makes a function measurable or a random variable, see Sigma-algebra#σ-algebra generated by a function

Join

Join may refer to:

Join (law), to include additional counts or additional defendants on an indictment

In mathematics:

Join (mathematics), a least upper bound of sets orders in lattice theory

Join (topology), an operation combining two topological spaces

Join (relational algebra), a binary operation on tuples corresponding to the relation join of SQL

Join (sigma algebra), a refinement of sigma algebras

Join (algebraic geometry), a union of lines between two varieties

Join (SQL), relational join, a binary operation on SQL and relational database tables

join (Unix), a Unix command similar to relational join

Join-calculus, a process calculus developed at INRIA for the design of distributed. programming languages

Joins (concurrency library), a concurrent computing API from Microsoft Research

Join Network Studio of NENU, a non-profit organization of Northeast Normal University

Joins.com, the website for South Korean newspaper JoongAng Ilbo

Kolmogorov's zero–one law

In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.

Tail events are defined in terms of infinite sequences of random variables. Suppose

is an infinite sequence of independent random variables (not necessarily identically distributed). Let be the σ-algebra generated by the . Then, a tail event is an event which is probabilistically independent of each finite subset of these random variables. (Note: belonging to implies that membership in is uniquely determined by the values of the but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event.

Intuitively, tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the are removed.

In many situations, it can be easy to apply Kolmogorov's zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine which of these two extreme values is the correct one.

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.The Lebesgue measure is often denoted by dx, but this should not be confused with the distinct notion of a volume form.

Measurable space

In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ {\displaystyle \sigma } -algebra on this set and provides information about the sets that will be measured.

Measure space

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. Measure spaces contain information about the underlying set, the subsets of said set that are feasible for measuring (the σ {\displaystyle \sigma } -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

Measure space should not be confused with the related measurable spaces.

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes.

Progressively measurable process

In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itô integrals.

Projection (measure theory)

In measure theory, projection map plays an important role in treating product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be measurable. Sometimes for some reasons product spaces are equipped with sigma-algebra different than the product sigma-algebra. In these cases the projections need not be measurable at all.

The projected set of a measurable set is called analytic set and need not be a measurable set. However, in some cases, either relatively to the product sigma-algebra or relatively to some other sigma-algebra, projected set of measurable set is indeed measurable.

Henri Lebesgue himself, one of the founders of measure theory, was mistaken about that fact. In a paper from 1905 he wrote that the projection of Borel set in the plane onto the real line is again a Borel set. The mathematician Mikhail Yakovlevich Suslin found that error about ten years later, and his following research has led to descriptive set theory. The fundamental mistake of Lebesgue was to think that projection commutes with decreasing intersection, while there are simple counterexamples to that.

Random element

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”The modern day usage of “random element” frequently assumes the space of values is a topological vector space, often a Banach or Hilbert space with a specified natural sigma algebra of subsets.

Sample space

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set").

For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}.

For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

Σ-Algebra of τ-past

The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probability theory.

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