Let (X, B, μ) be a measure space. Prove the following: If{Ei} is a countable collection of sets in B with μ(E1) < ∞and Ei ⊇ Ei+1 for i = 1, 2,..., thenμ ∞i=1Ei= limn→∞ μ(En). (8) Let (X, B, u) be a measure space. Prove the following: If{E} is a countable collection of sets in B with u(E1) < 00and E, 2 E+1 for i 1,2,..., then= lim u(En).

Answered: Image /qna-images/answer/b37148af-5552-45d8-958c-f5ede573f0f7.jpg

Answered: (8) Let (X, B, µ) be a measure space.…

(8) Let (X, B, µ) be a measure space. Prove the following: If {E} is a countable collection of sets in B with u(E1) < 0 and E, 2 E+1 for i 1,2,..., then E, = lim u(En).

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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Question

Let (X, B, μ) be a measure space. Prove the following: If
{Ei} is a countable collection of sets in B with μ(E1) < ∞
and Ei ⊇ Ei+1 for i = 1, 2,..., then
μ

∞
i=1
Ei

= limn→∞ μ(En).

(8) Let (X, B, u) be a measure space. Prove the following: If
{E} is a countable collection of sets in B with u(E1) < 00
and E, 2 E+1 for i 1,2,..., then
= lim u(En).

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