Prove the following 1) If S is o-algebra and FC S, then σ(F) ≤ S2) o(F) = F ⇒F is a o-algebra3) σ(F₁) = σ (F₂) ↔ F₁ ≤ 0(F₂) ^ F₂ ≤ 0 (F₁).4) Let Fc = {A: A E F}. Then o(F) = o(F)

Answered: 1) If S is o-algebra and FCS, then σ(F)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

100%

Prove the following

1) If S is o-algebra and FC S, then σ(F) ≤ S
2) o(F) = F ⇒F is a o-algebra
3) σ(F₁) = σ (F₂) ↔ F₁ ≤ 0(F₂) ^ F₂ ≤ 0 (F₁).
4) Let Fc = {A: A E F}. Then o(F) = o(F)

Expert Solution

Step 1

Disclaimer: Since you have asked multiple question, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.

What is Sigma-Algebra:

A $σ$ -algebra (also $σ$ -field) on a set X is a nonempty collection of X subsets closed under complement, countable unions, and countable crossings in mathematical analysis and probability theory. X and are referred to as a measurable space. The set algebras, which the $σ$ -algebras are a subset of, merely require that their elements be closed under the union or intersection of finitely many subsets, a lesser requirement.

Given:

It is given that $S$ is a $σ - Algebra$ and $F \subseteq S$ .

To Prove:

We prove that $σ (F) \subseteq S$ .