What is the point of Sigma algebra?
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.
Are sigma algebras unique?
Let G⊆P(X) be a collection of subsets of X. Then σ(G), the σ-algebra generated by G, exists and is unique.
Is Sigma algebra a set?
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.
How do you make the smallest sigma algebra?
To obtain the smallest σ-algebra containing it, all you need to do is add the missing sets that make it a σ-algebra (instead of just being a set). What this means is that you want to add all sets so that the resulting set is closed with respect to taking complements and union.
Is sigma algebra a set?
Why is it called a sigma-algebra?
In the words “σ-ring”,”σ-algebra” the prefix “σ-…” indicates that the system of sets considered is closed with respect to the formation of denumerable unions. Here the letter σ is to remind one of “Summe”[sum]; earlier one refered to the union of two sets as their sum (see for example F. Hausdorff 1, p.
How do you make the smallest sigma-algebra?
Is Sigma countably infinite?
By the same construction as above, they can be mapped to the one-element sets of natural numbers, which means that their closure is uncountably infinite. Therefore, the assumption that countably infinite sigma algebras exist is false.
What is the difference between a sigma algebra and an algebra?
$\\begingroup$ An algebra is a collection of subsets closed under finite unions and intersections. A sigma algebra is a collection closed under countable unions and intersections.
How is a sigma algebra different from a topology?
A sigma algebra is a collection closed under countable unions and intersections. In either case, complements are also included. A topology starts with the idea that certain sets are open. The requirement is that the collection include all unions.
Which is the best definition of a σ algebra?
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. 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.
Can a sigma algebra be closed under an infinite intersection?
Finally – algebras and sigma algebras are collections of sets. To be closed under finite intersections means that taking any number of finite intersections of elements of the algebra yields an element (another set) that is in the algebra. But maybe this isn’t true for an infinite intersection, etc.