What is generated Sigma algebra?
The generated σ-algebra or generated σ-field refers to. The smallest σ-algebra that contains a given family of sets, see Generated σ-algebra (by sets) The smallest σ-algebra that makes a function measurable or a random variable, see Sigma-algebra#σ-algebra generated by a function.
How do you show something is a sigma algebra?
1.1. A set of sets A is a σ-algebra if and only if (i) Ω∈A, (ii) A∈A implies Ac∈A, and (iii) if An∈A for n∈N then ∪nAn∈A.
What is trivial sigma algebra?
Definition 1 A collection F of subsets of Ω is called a σ-algebra (or σ-field) if the following hold. F = 2Ω = {A|A ⊆ Ω}, the power set of Ω. 2. F = {∅,Ω}, the trivial σ-algebra.
Which is an example of a sigma-algebra induced by a random variable?
For the reason that we are dealing with preimages of the random variable X, we call Σ the sigma-algebra induced by the random variable X. Here is an extreme example: consider a constant random variable X, that is, X(ω) ≡ α. Then X − 1(B), B ∈ B(R) equals either Ω or ∅ depending on whether α ∈ B.
Which is the smallest algebra that makes X a random variable?
Σ is in fact the smallest sigma-algebra that makes X a random variable as all other sigma-algebras of that kind would at the very least include Σ. For the reason that we are dealing with preimages of the random variable X, we call Σ the sigma-algebra induced by the random variable X.
How to show that σ is a sigma algebra?
Using the properties of preimages, it is not too difficult to show that Σ is a sigma-algebra. It also follows immediately that Σ ⊂ A, hence Σ is a sub-sigma-algebra. Further, by the definitions it is easy to see that the mapping X: (Ω, Σ) → (R, B(R)) is measurable.
Is the sigma algebra trivial in cross validated?
The sigma-algebra thus generated is trivial and as such, it is definitely included in A. Hope this helps. Thanks for contributing an answer to Cross Validated!