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, is closed under complement, and is closed under countable unions and countable intersections. The pair (X, Σ) is called a measurable space.

What is sigma-algebra in stochastic process?

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, is closed under complement, and is closed under countable unions and countable intersections. The pair (X, Σ) is called a measurable space.

What is sigma-algebra examples?

Definition The σ-algebra generated by Ω, denoted Σ, is the collection of possible events from the experiment at hand. Definition The σ-algebra generated by Ω, denoted Σ, is the collection of possible events from the experiment at hand. Example: We have an experiment with Ω = {1, 2}. Then, Σ = {{Φ},{1},{2},{1,2}}.

How do you show that a set is a sigma-field?

Proving a set is a Sigma Algebra

1. If we let A=X, then clearly Y∈B.
2. Suppose G∈B. Show Gc∈B.
3. So G=A∩Y for some A∈D.

Why are measures defined on sigma algebras?

If your probability depends on volume, by changing the volume of the set you will also change probabilities. This means that no event can have a single probability assigned to it. This necessitates sigma algebra, which allows us to define measurable sets and probabilities.

Why is it called sigma-algebra?

The letters σ and δ are often given as Greek abbreviations of German words: σ as S in Summe for sum (in the sense of sum of sets, that is, union) and δ as D in Durchschnitt for intersection, both countable.

Why sigma-algebra is needed?

Sigma algebra is necessary in order for us to be able to consider subsets of the real numbers of actual events. In other words, the sets need to be well defined, under the conditions of countable unions and countable intersections, for it to have probabilities assigned to it.

Why is it called a sigma algebra?

Is Sigma field and Sigma algebra the same?

In general, the term σ-algebra is used by people doing pure analysis, and the term σ-field is used by probability theorists. They are the same thing, however.

What are stochastic processes?

A stochastic process means that one has a system for which there are observations at certain times, and that the outcome, that is, the observed value at each time is a random variable.