Sigma-algebraFrom Wikipedia, the free encyclopedia

Chapter 1

Join (sigma algebra)

In mathematics, the join of two sigma algebras over the same set X is the coarsest sigma algebra containing both.[1]

1.1 References[1] Peter Walters, An Introduction to Ergodic Theory (1982) Springer Verlag, ISBN 0-387-90599-5

2

Chapter 2

Separable sigma algebra

In mathematics, σ-algebras are usually studied in the context of measure theory. A separable σ-algebra (or sepa-rable σ-field) is a σ-algebra F which is a separable space when considered as a metric space with metric ρ(A,B) =µ(AB) for A,B ∈ F and a given measure µ (and with being the symmetric difference operator). Note thatany 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 notcountably 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 distancebetween two sets is defined as the measure of the symmetric difference of the two sets. Note that the symmetricdifference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a truemetric. 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 canbe shown that the corresponding metric space is, too.

3

Chapter 3

Sigma additivity

In mathematics, additivity and sigma additivity (also called countable additivity) of a function defined on subsetsof a given set are abstractions of the intuitive properties of size (length, area, volume) of a set.

3.1 Additive (or finitely additive) set functions

Let µ be a function defined on an algebra of sets A with values in [−∞, +∞] (see the extended real number line).The function µ is called additive, or finitely additive, if, whenever A and B are disjoint sets in A , one has

µ(A ∪B) = µ(A) + µ(B).

(A consequence of this is that an additive function cannot take both −∞ and +∞ as values, for the expression ∞ − ∞is undefined.)One can prove by mathematical induction that an additive function satisfies

µ

(N∪

n=1

An

)=

N∑n=1

µ(An)

for any A1, A2, . . . , AN disjoint sets in A .

3.2 σ-additive set functions

Suppose that A is a σ-algebra. If for any sequence A1, A2, . . . , An, . . . of disjoint sets in A , one has

µ (∪∞

n=1 An) =∑∞

n=1 µ(An) ,

we say that μ is countably additive or σ-additive.Any σ-additive function is additive but not vice versa, as shown below.

3.3 τ-additive set functions

Suppose that in addition to a sigma algebra A , we have a topology τ. If for any directed family of measurable opensets G ⊆ A ∩τ,

µ (∪

G) = supG∈G µ(G) ,

we say that μ is τ-additive. In particular, if μ is inner regular then it is τ-additive.[1]

4

3.4. PROPERTIES 5

3.4 Properties

3.4.1 Basic properties

Useful properties of an additive function μ include the following:

1. Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain.

2. If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).

3. If A ⊆ B and μ(B) − μ(A) is defined, then μ(B \ A) = μ(B) − μ(A).

4. Given A and B, μ(A ∪ B) + μ(A ∩ B) = μ(A) + μ(B).

3.5 Examples

An example of a σ-additive function is the function μ defined over the power set of the real numbers, such that

µ(A) =

1 if 0 ∈ A

0 if 0 /∈ A.

If A1, A2, . . . , An, . . . is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, orprecisely one of them does. In either case, the equality

µ

( ∞∪n=1

An

)=

∞∑n=1

µ(An)

holds.See measure and signed measure for more examples of σ-additive functions.

3.5.1 An additive function which is not σ-additive

An example of an additive function which is not σ-additive is obtained by considering μ, defined over the Lebesguesets of the real numbers by the formula

µ(A) = limk→∞

1

k· λ (A ∩ (0, k)) ,

where λ denotes the Lebesgue measure and lim the Banach limit.One can check that this function is additive by using the linearity of the limit. That this function is not σ-additivefollows by considering the sequence of disjoint sets

An = [n, n+ 1)

for n=0, 1, 2, ... The union of these sets is the positive reals, and μ applied to the union is then one, while μ appliedto any of the individual sets is zero, so the sum of μ(An) is also zero, which proves the counterexample.

3.6 Generalizations

One may define additive functions with values in any additive monoid (for example any group or more commonly avector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on thatset. For example, spectral measures are sigma-additive functions with values in a Banach algebra. Another example,also from quantum mechanics, is the positive operator-valued measure.

6 CHAPTER 3. SIGMA ADDITIVITY

3.7 See also• signed measure

• measure (mathematics)

• additive map

• subadditive function

• σ-finite measure

• Hahn–Kolmogorov theorem

This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

3.8 References[1] D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.

Chapter 4

Sigma-algebra

"Σ-algebra” redirects here. For an algebraic structure admitting a given signature Σ of operations, see Universal al-gebra.

In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a setX is a collection Σ of subsets of X that is closed under countable-fold set operations (complement, union of countablymany sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitelymany set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations.The pair (X, Σ) is also a field of sets, called a measurable space.The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for whicha given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as thefoundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events whichcan be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.In statistics, (sub) σ-algebras are needed for a formal mathematical definition of sufficient statistic,[1] particularlywhen 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.However, 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 (includingthe 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 addingin all countable unions, countable intersections, and relative complements and continuing this process (by transfiniteiteration through all countable ordinals) until the relevant closure properties are achieved (a construction known asthe Borel hierarchy).

4.1 Motivation

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

4.1.1 Measure

Ameasure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as makingprecise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of theirindividual 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 theaxiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the realline, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead asmaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

7

8 CHAPTER 4. SIGMA-ALGEBRA

operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable setand the countable union of measurable sets is a measurable set. Non-empty collections of sets with these propertiesare called σ-algebras.

4.1.2 Limits of sets

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

• The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim supn→∞

An =∞∩

n=1

∞∪m=n

Am.

• The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

lim infn→∞

An =∞∪

n=1

∞∩m=n

Am.

• If, in fact,

lim infn→∞

An = lim supn→∞

An

then the limn→∞ An exists as that common set.

4.1.3 Sub σ-algebras

In much of probability, especially when conditional expectation is involved, one is concerned with sets that representonly part of all the possible information that can be observed. This partial information can be characterized with asmaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only toand determined only by the partial information. A simple example suffices to illustrate this idea.Imagine you are playing 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 :

Ω = H,T∞ = (x1, x2, x3, . . . ) : xi ∈ H,T, i ≥ 1

However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the nextflip. 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

Gn = A× H,T∞ : A ⊂ H,Tn

Observe that then

G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G∞

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

4.2. DEFINITION AND PROPERTIES 9

4.2 Definition and properties

4.2.1 Definition

Let X be some set, and let 2X represent its power set. Then a subset Σ ⊂ 2X is called a σ-algebra if it satisfies thefollowing three properties:[2]

1. X is in Σ.

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 = A1 ∪ A2 ∪ A3 ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying DeMorgan’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, by (3) it follows as well that X, ∅ is the smallest possible σ-algebra.Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebraover X, is called a measurable space. A function between two measurable spaces is called a measurable function ifthe preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with themeasurable 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).

4.2.2 Dynkin’s π-λ theorem

This theorem (or the related monotone class theorem) is an essential tool for proving many results about propertiesof 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 Σ that is closed under finitely many intersections, anda Dynkin system (or λ-system) D is a collection of subsets of Σ that contains Σ and is closed undercomplement 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 verifythat all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of allsubsets with the property is a Dynkin system can also be straightforward. Dynkin’s π-λ Theorem then implies thatall 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 orintegrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integraltypically associated with computing the probability:

P(X ∈ A) =∫AF (dx) for all A in the Borel σ-algebra on R,

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

4.2.3 Combining σ-algebras

Suppose Σα : α ∈ A 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 oftenis denoted by:

10 CHAPTER 4. SIGMA-ALGEBRA

∧α∈A

Σα.

Sketch of Proof: Let Σ∗ denote the intersection. Since X is in every Σα, Σ∗ is not empty. Closure undercomplement and countable unions for every Σα implies the same must be true for Σ∗. Therefore, Σ∗ is aσ-algebra.

• The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates aσ-algebra known as the join which typically is denoted

∨α∈A

Σα = σ

( ∪α∈A

Σα

).

A π-system that generates the join is

P =

n∩

i=1

Ai : Ai ∈ Σαi , αi ∈ A, n ≥ 1

.

Sketch of Proof: By the case n = 1, it is seen that each Σα ⊂ P , so∪α∈A

Σα ⊂ P.

This implies

σ

( ∪α∈A

Σα

)⊂ σ(P)

by the definition of a σ-algebra generated by a collection of subsets. On the other hand,

P ⊂ σ

( ∪α∈A

Σα

)

which, by Dynkin’s π-λ theorem, implies

σ(P) ⊂ σ

( ∪α∈A

Σα

).

4.2.4 σ-algebras for subspaces

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

• The collection Y ∩ B: B ∈ Σ is a σ-algebra of subsets of Y.

• Suppose (Y, Λ) is a measurable space. The collection A ⊂ X : A ∩ Y ∈ Λ is a σ-algebra of subsets of X.

4.2.5 Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[3] A σ-ring need not be a σ-algebra, as for examplemeasurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real linehas infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takesmeasurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtainedby their countable union yet its measure is not finite.

4.3. EXAMPLES 11

4.2.6 Typographic note

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

4.3 Examples

4.3.1 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 (whichis distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by thesingletons 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.

4.3.2 Stopping time sigma-algebras

A stopping time τ can define a σ -algebraFτ , the so-called stopping time sigma-algebra, which in a filtered probabilityspace describes the information up to the random time τ in the sense that, if the filtered probability space is interpretedas a random experiment, the maximum information that can be found out about the experiment from arbitrarily oftenrepeating it until the time τ is Fτ .[4]

4.4 σ-algebras generated by families of sets

4.4.1 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 setin F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containingF. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated byF.For a simple example, consider the set X = 1, 2, 3. Then the σ-algebra generated by the single subset 1 isσ(1) = ∅, 1, 2, 3, 1, 2, 3. By an abuse of notation, when a collection of subsets contains only oneelement, A, one may write σ(A) instead of σ(A); in the prior example σ(1) instead of σ(1). Indeed, usingσ(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.

4.4.2 σ-algebra generated by a function

If f is a function from a set X to a set Y and B is a σ-algebra of subsets of Y, then the σ-algebra generated by thefunction f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B. i.e.

σ(f) = f−1(S) |S ∈ B.

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) isa subset of Σ.

12 CHAPTER 4. SIGMA-ALGEBRA

One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topologicalspace 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 inter-vals/rectangles in Rn:

σ(f) = σ(f−1((a1, b1]× · · · × (an, bn]) : ai, bi ∈ R

).

A useful property is the following. Assume f is a measurable map from (X, ΣX) to (S, ΣS) and g is a measurable mapfrom (X, ΣX) to (T, ΣT). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂σ(g). If S is finite or countably infinite or if (S, ΣS) is a standard Borel space (e.g., a separable complete metric spacewith its associated Borel sets) then the converse is also true.[5] Examples of standard Borel spaces include Rn with itsBorel sets and R∞ with the cylinder σ-algebra described below.

4.4.3 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-trivialexample 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-algebracontains more sets than the Borel σ-algebra onRn and is preferred in integration theory, as it gives a complete measurespace.

4.4.4 Product σ-algebra

Let (X1,Σ1) and (X2,Σ2) be two measurable spaces. The σ-algebra for the corresponding product spaceX1 ×X2

is called the product σ-algebra and is defined by

Σ1 × Σ2 = σ(B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2).

Observe that B1 ×B2 : B1 ∈ Σ1, B2 ∈ Σ2 is a π-system.The Borel σ-algebra for Rn is generated by half-infinite rectangles and by finite rectangles. For example,

B(Rn) = σ ((−∞, b1]× · · · × (−∞, bn] : bi ∈ R) = σ ((a1, b1]× · · · × (an, bn] : ai, bi ∈ R) .

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

4.4.5 σ-algebra generated by cylinder sets

Suppose

X ⊂ RT = f : f(t) ∈ R, t ∈ T

is a set of real-valued functions. Let B(R) denote the Borel subsets ofR. A cylinder subset of X is a finitely restrictedset defined as

Ct1,...,tn(B1, . . . , Bn) = f ∈ X : f(ti) ∈ Bi, 1 ≤ i ≤ n.

Each

Ct1,...,tn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n

4.5. SEE ALSO 13

is a π-system that generates a σ-algebra Σt1,...,tn . Then the family of subsets

FX =

∞∪n=1

∪ti∈T,i≤n

Σt1,...,tn

is an algebra that generates the cylinder σ-algebra for X. This σ-algebra is a subalgebra of the Borel σ-algebradetermined by the product topology of RT restricted to X.An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In thiscase, it suffices to consider the cylinder sets

Cn(B1, . . . , Bn) = (B1 × · · · ×Bn × R∞) ∩X = (x1, x2, . . . , xn, xn+1, . . . ) ∈ X : xi ∈ Bi, 1 ≤ i ≤ n,

for which

Σn = σ(Cn(B1, . . . , Bn) : Bi ∈ B(R), 1 ≤ i ≤ n)

is a non-decreasing sequence of σ-algebras.

4.4.6 σ-algebra generated by random variable or vector

Suppose (Ω,Σ,P) is a probability space. If Y : Ω → Rn 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(Y ) = Y −1(A) : A ∈ B(Rn).

4.4.7 σ-algebra generated by a stochastic process

Suppose (Ω,Σ,P) is a probability space and RT is the set of real-valued functions on T . If Y : Ω → X ⊂ RT ismeasurable with respect to the cylinder σ-algebra σ(FX) (see above) for X then Y is called a stochastic process orrandom process. The σ-algebra generated by Y is

σ(Y ) =Y −1(A) : A ∈ σ(FX)

= σ(Y −1(A) : A ∈ FX),

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

4.5 See also

• Join (sigma algebra)

• Measurable function

• Sample space

• Separable sigma algebra

• Sigma ring

• Sigma additivity

14 CHAPTER 4. SIGMA-ALGEBRA

4.6 References[1] Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1118122372.

[2] Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

[3] Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 9780470317952.

[4] Fischer, Tom (2013). “On simple representations of stopping times and stopping time sigma-algebras”. Statistics andProbability Letters 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.

[5] Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

4.7 External links• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Sigma Algebra from PlanetMath.

Chapter 5

Sigma-ring

Inmathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countableunion and relative complementation.

5.1 Formal definition

LetR be a nonempty collection of sets. ThenR is a σ-ring if:

1.∪∞

n=1 An ∈ R if An ∈ R for all n ∈ N

2. A \B ∈ R if A,B ∈ R

5.2 Properties

From these two properties we immediately see that

∩∞n=1 An ∈ R if An ∈ R for all n ∈ N

This is simply because ∩∞n=1An = A1 \ ∪∞

n=1(A1 \An) .

5.3 Similar concepts

If the first property is weakened to closure under finite union (i.e.,A∪B ∈ RwheneverA,B ∈ R ) but not countableunion, thenR is a ring but not a σ-ring.

5.4 Uses

σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one doesnot wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be aσ-field.A σ-ring R that is a collection of subsets of X induces a σ-field for X . Define A to be the collection of all subsetsof X that are elements ofR or whose complements are elements ofR . Then A is a σ-field over the set X . In factA is the minimal σ-field containingR since it must be contained in every σ-field containingR .

15

16 CHAPTER 5. SIGMA-RING

5.5 See also• Delta ring

• Ring of sets

• Sigma field

5.6 References• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-ringsin development of Lebesgue theory.

5.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 17

5.7 Text and image sources, contributors, and licenses

5.7.1 Text• Join (sigma algebra) Source: https://en.wikipedia.org/wiki/Join_(sigma_algebra)?oldid=587405272 Contributors: Tsirel, Linas andAnonymous: 1

• Separable sigma algebra Source: https://en.wikipedia.org/wiki/Separable_sigma_algebra?oldid=654658454Contributors: Tobias Berge-mann, Tsirel, Oleg Alexandrov, Salix alba, CBM, Thijs!bot, Addbot, Topology Expert, Yobot, AnomieBOT, Erik9bot, RedBot, Zfeinst,Qetuth, Khazar2 and Anonymous: 3

• Sigma additivity Source: https://en.wikipedia.org/wiki/Sigma_additivity?oldid=670579000 Contributors: Michael Hardy, Chinju, An-dres, Charles Matthews, Fibonacci, MathMartin, Vivacissamamente, Paul August, Rgdboer, Blotwell, Oleg Alexandrov, Linas, Salix alba,צבי ,דניאל A. Pichler, WISo, TM-77, R'n'B, Mikemtha, Daniele.tampieri, Dingenis, Addbot, SamatBot, Luckas-bot, Yobot, RandomDS-devel, 777sms, John of Reading, ZéroBot, Quondum, WoldaCZ and Anonymous: 15

• Sigma-algebra Source: https://en.wikipedia.org/wiki/Sigma-algebra?oldid=686336064Contributors: AxelBoldt, Zundark, Tarquin, Iwn-bap, Miguel~enwiki, Michael Hardy, Chinju, Karada, Stevan White, Charles Matthews, Dysprosia, Vrable, AndrewKepert, Fibonacci,Robbot, Romanm, Ruakh, Giftlite, Lethe, MathKnight, Mboverload, Gubbubu, Gauss, Barnaby dawson, Vivacissamamente, WilliamElliot, ArnoldReinhold, Paul August, Bender235, Zaslav, Elwikipedista~enwiki, MisterSheik, EmilJ, SgtThroat, Jung dalglish, Tsirel,Passw0rd, Msh210, Jheald, Cmapm, Ultramarine, Oleg Alexandrov, Linas, Graham87, Salix alba, FlaBot, Mathbot, Jrtayloriv, Chobot,Jayme, YurikBot, Lucinos~enwiki, Archelon, Trovatore, Mindthief, Solstag, Crasshopper, Dinno~enwiki, Nielses, SmackBot, Melchoir,JanusDC, Object01, Dan Hoey, MalafayaBot, RayAYang, Nbarth, DHN-bot~enwiki, Javalenok, Gala.martin, Henning Makholm, Lam-biam, Dbtfz, Jim.belk, Mets501, Stotr~enwiki, Madmath789, CRGreathouse, CBM, David Cooke, Mct mht, Blaisorblade, Xanthar-ius, , Thijs!bot, Escarbot, Keith111, Forgetfulfunctor, Quentar~enwiki, MSBOT, Magioladitis, RogierBrussee, Paartha, Joeabauer,Hippasus, Policron, Cerberus0, Digby Tantrum, Jmath666, Alfredo J. Herrera Lago, StevenJohnston, Ocsenave, Tcamps42, SieBot, Mel-combe, MicoFilós~enwiki, Andrewbt, The Thing That Should Not Be, Mild Bill Hiccup, BearMachine, 1ForTheMoney, DumZiBoT, Ad-dbot, Luckas-bot, Yobot, Li3939108, Amirmath, Godvjrt, Xqbot, RibotBOT, Charvest, FrescoBot, BrideOfKripkenstein, AstaBOTh15,Stpasha, RedBot, Soumyakundu, Wikiborg4711, Stj6, TjBot, Max139, KHamsun, Rafi5749, ClueBot NG, Thegamer 298, QuarkyPi,Brad7777, AntanO, Shalapaws, Crawfoal, Dexbot, Y256, Jochen Burghardt, A koyee314, Limit-theorem, Mark viking, NumSPDE,Moyaccercchi, Killaman slaughtermaster, DarenCline, 5D2262B74 and Anonymous: 82

• Sigma-ring Source: https://en.wikipedia.org/wiki/Sigma-ring?oldid=667498132 Contributors: Charles Matthews, Touriste, Salix alba,Maxbaroi, Konradek, Keith111, Addbot, Luckas-bot, Xqbot, ZéroBot, Zephyrus Tavvier, Jim Sukwutput, Brirush, DarenCline, Gana-tuiyop and Anonymous: 3

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