Question:Let N be a non-empty set and B a o-ring on N. Let A E B and define{B € B; B C A}. Show that Ba is a o-algebra on A.BALet N be a non-empty set and let P(N) denote its power set.Definition. A non-empty subset B of P(N) is called a ring (of sets) onN if the following hold:(i) If A, B E B, then A\ Be B.(ii) If A, B e B, then AU B e B.A ring B is called an algebra, if N E B.A ring B is called a o-ring(iii) if U A, E B for every sequence (An) in B.nƐNA o-ring B is called a o-algebra if N E B. The sets in a o-ring are calledmeasurable sets.

We have, Ω, a non-empty set and, B is a σ-ring on Ω. Then, we have A∈B, and define BA={B∈B; B⊆A}. We have to show that BA is a σ-algebra on A. Now, let A be a set. A set BA is a σ-algebra on A, if BA is a collection of subsets of A, such thati) A∈BAii) if, G,H ∈BA, then G\H∈BAiii) if (Gn) is any sequence in BA, then ∪nGn∈BA. i) Since A∈B, and A⊆A, we can say that A∈BA. ii) Let G,H∈BA⇒G,H∈B such that G⊆A, H⊆A. We know that B is a σ-ring, therefore, for G,H ∈B, G\H∈B. Also by property of sets, for G⊆A, H⊆A, we have G\H⊆AThus, for G,H∈BA, we get G\H∈BA. iii) Let (Gn) be a sequence in BA, then by definition (Gn)∈B, Gn⊆A, ∀n. Since B is a σ-ring, for sequence (Gn)∈B, we have ∪nGn∈B. Now, A itself is an element of B, and Gn⊆A,∀n, then from the property, if A, B∈B,A∪B∈B, we get that∪nGn⊆A. Thus, ∪nGn∈BA Since we have proved all the conditions, we can say that BA is a σ-algebra on A.

Answered: Let N be a non-empty set and B a o-ring…

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Advanced Math

Let N be a non-empty set and B a o-ring on N. Let A e B and define BA = {B e B; BC A}. Show that Ba is a o-algebra on A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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