Measurable space

In mathematical analysis, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a set is a collection of subsets satisfying certain properties. 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 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.

The definition is that a σ-algebra over a set X is a nonempty collection Σ of subsets of X that is closed under the complement and countable unions of its members and contains X itself.

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.

If X = {a, b, c, d}, one possible sigma algebra on X is Σ = {∅, {a, b}, {c, d}, {a, b, c, d}}, where ∅ is the empty set. 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 until the relevant closure properties are achieved (a construction known as the Borel σ-algebra [see Borel set]).

Motivation

A measure on X is a function which 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.

The collection of subsets of X that form the σ-algebra is usually denoted by Σ, the capital Greek letter sigma. The pair (X, Σ) is an algebra of sets and also a field of sets, called a measurable space. If the subsets of X in Σ correspond to numbers in elementary algebra, then the two set operations union (symbol ∪) and intersection (∩) correspond to addition and multiplication. The collection of sets Σ is completed to include countably infinite operations.

Definition and properties

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 the following three properties:[1]

  1. Σ is non-empty: There is at least one A X 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 = A1A2A3 ∪ … .

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

It also follows that the set X itself and the empty set are both in Σ, since by (1) Σ is non-empty, so some particular A ∈ Σ may be chosen, and by (2), X \ A is also in Σ. By (3) A ∪ (X \ A) = X is in Σ. And finally, since X is in Σ, (2) asserts that its complement, the empty set, is also in Σ.

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, ∞].

Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X. 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 \scriptstyle(X,\,\mathcal{F}) or \scriptstyle(X,\,\mathfrak{F}). This is handy to avoid situations where the letter Σ may be confused for the summation operator.

σ-algebra generated by a family of sets

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). This σ-algebra is denoted σ(F) and called the σ-algebra generated by F.

To see that such a σ-algebra always exists, let

Φ ={ E ⊂ 2X : E is a σ -algebra which contains F }.

The σ-algebra generated by F will therefore be the smallest element in Φ. Indeed, such a smallest element exists: First, Φ is not empty because the power set 2X is in Φ. Consequently, let σ* denote the intersection of all elements in Φ. This intersection will be nonempty, moreover, it will also be a σ algebra, because each element in Φ is a σ-algebra and the arbitrary intersection of σ-algebras is a σ-algebra (observe that if every element in Φ has the three properties of a σ-algebra, then the intersection of Φ will as well). Because each element in Φ contains F, the intersection σ* will also contain F. Hence, because σ* is a σ-algebra which contains F, σ* is in Φ, and because it is the intersection of all sets in Φ, σ* is indeed the smallest set in Φ by definition, which in turn implies that σ* = σ(F), the σ-algebra generated by F.

For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the single subset {1} is σ() = {∅, {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(). Also when that subset contains only one element, a, one may write σ(a) instead of σ(A)=σ({a}); in the prior example σ(1) instead of σ({1}).

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 the function f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B.

Clearly, 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 are the Borel sets on Y.

Examples

Let X be any set, then the following are σ-algebras over X:

  • 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 of subsets of X which are countable or whose complements are countable (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.
  • If {Σλ} is a family of σ-algebras over X indexed by λ then the intersection of all Σλ's is a σ-algebra over X.

Examples for generated 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, see the Vitali set.

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.

See also

References

External links

  • Template:Springer
  • Sigma Algebra from PlanetMath.

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