How to generate a family of sigma algebras?

How to generate a family of sigma algebras?

Now take the sets that are common to all such sigma algebras. That gives the sigma algebra generated by our family. In this case this turns out to be exactly countable sets and their complements.

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.

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.

Can a σ algebra be generated from a semiring?

All σ-algebras are algebras, and all algebras are semi-rings. Thus, if we require a set to be a semiring, it is sufficient to show instead that it is a σ-algebra or algebra. Sigma algebras can be generated from arbitrary sets. This will be useful in developing the probability space.

Which is a sigma algebra containing all countable sets?

One sigma algebra containing these sets is the power set. Another one is the family of all countable sets and their complements. There are many others too. Now take the sets that are common to all such sigma algebras. That gives the sigma algebra generated by our family.

How do you find the smallest sigma algebra?

Further, we can find the smallest sigma algebra by intersecting all sigma algebras containing A, as the intersection of sigma algebras is also a sigma algebra. The last part is the one I don’t understand and confuses me.

Why does a sigma algebra contain all singletons?

Reason: Any sigma algebra that contains singletons must contain all countable sets (since they are countable unions of singletons). It must also contain their complements. Hence the smallest sigma algebra containig all singeltons is exactly the family of all countable sets and their complements.