Sigma-algebra

A σ - algebra (also σ - algebra, Sigma body or Borel amount of body ) is a key concept in measure theory, where σ - algebras occur as domains of definition for dimensions. A σ - algebra is a set-theoretic structure that identifies a set system on a fixed base amount, which contains the initial amount and is complete with respect to the complementation and countable unions. In the stochastics, which is based on measure theory, play σ - algebras as event spaces play an important role as systems of quantities that are interpreted as events.

• 7.1 example

Definition

As σ - algebra is called in mathematics a lot of system ( called the power set ), ie a set of subsets of the universal set that satisfies the following conditions:

Notes

• Follows from the conditions 1 and 2, that will always contain the complement of, that is, the empty set. Because of Property 2 one can ask for in property 1 alternatively.
• If one chooses in Condition 3, the quantities for all, it follows that the finite union of sets is included.
• If for every natural number, it follows from the De Morgan 's laws and the conditions 2 and 3 show that the average amount in is because

• If one chooses for all, it follows that the intersection of finitely many sets is included. A σ - algebra is thus completed over finite and countably infinite averages.
• Are in and out, as is also. So is closed under set difference.
• Furthermore, each σ - algebra is in particular also a Dynkin system.
• Is a finite σ - algebra, so there is always a non-negative integer, which means that the cardinality of is a power of two.

Examples

• For any quantity is the smallest and the power set of the largest possible σ - algebra as the basic quantity.
• For any set and subset is the smallest σ - algebra containing.
• For any topological space, the σ - algebra of Borel subsets of, inter alia, contains all open and closed subsets of exist.
• The σ - algebra of Borel subsets of the real numbers contains, among other things, all intervals.
• About a basic amount of the quantity system is a σ - algebra. Is this uncountable, so is a function if and only measurable when it is constant on the complement of a countable set.
• Are and any two quantities, a σ - algebra and a picture. Then a σ -algebra in.

Importance

σ - algebras form the starting point for the definition of Maßraums and the probability space. Demonstrates the Banach - Tarski paradox that on uncountable sets that formed by the power set σ - algebra as a basis for determining the volumes may be too large and the consideration of other σ - algebras is mathematically necessary. In the theory of stochastic processes, in particular in stochastic financial mathematics and to a time in principle observable information is described by a σ - algebra, which, so a time- ascending family of σ - algebras leads to the concept of filtration. Filtrations are essential for the general theory of stochastic integration; Integrand (ie actuarial trading strategies ) may depend at a time only on the information to (but excluding ); in particular, they may not " look into the future ".

σ - operator

1 For an arbitrary subset of the power set of the operator is defined as

In which

Since the intersection of a family of σ - algebras (over the same ground set ) is a σ - algebra again, so is the smallest σ - algebra which includes.

The operator complies with the fundamental properties of a shell operator:

• , So the operator is extensive.
• Applies, as is also ( or monotony isotonicity ).
• It is ( idempotency ).

Is referred to as the generated by σ algebra, this is producer σ algebra.

In many cases, the elements not explicitly specified (see, for example, Borel hierarchy). A frequently used method of proof for statements that apply to all elements of, is the principle of good quantities.

2 If functions of measurement in rooms, so is

The smallest σ algebra over in respect of which that are measurable. It is designated by σ generated algebra. The same is true for arbitrary index sets instead.

Product σ - algebra

For a family of measuring rooms there is a smallest σ - algebra on the ( Cartesian ) product of, so that all projections are measurable on, it is therefore

Which are the projections of the individual. is called the product σ - algebra of ( also Kolmogorowsche σ - algebra). The couple

Forming a measuring chamber, which is also referred to as a measurable product in the family.

If, as one often writes or place.

If for all, we used partially also the notation for the corresponding product σ - algebra.

You can also take the help of producers of feature:

Is countable (or finite ), then

In which

The product of the family. Note that the product of two σ - algebras and generally no σ -algebra. However, a semi-ring and in particular stable.

Examples

• Be and σ - algebras. Then the corresponding product σ - algebra:

Here is the product of σ - algebras and.

• The Borel σ - algebra on is equal to the product σ - algebra, it is thus:

Application

Product σ - algebras are the basis for the theory of product measures, in turn, form the basis for the general Fubini's theorem.

For the stochastics are product - σ - algebras of fundamental importance in order to make statements about the existence of product probability measures and product probability spaces. These are, firstly, important to describe multi-stage random experiments, and on the other, fundamental to the theory of stochastic processes.

Trace σ - algebra

For the amount of system is called a trace of in or track - σ - algebra of about. One can show that the trace of a σ - algebra in again (but with the base amount ) is what the name " trace σ - algebra " justifies.

Example

• Be a corresponding σ - algebra, and so is the trace σ - algebra of about.

Terminals σ - algebra

The terminal σ - algebra is a special σ - algebra, which is in probability theory of meaning.