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1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = 0 is in M. Since M only contains o and X then øUX = X € M. Thus, M is ao-algebra because the three conditions are satisfied. This means that (X, M) is a measurable space. Exercise 1. Use the same proving technique used in showing T = p(X) is a topology to show that M = p(X) is ao-algebra.

1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = 0 is in M. Since M only contains o and X then øUX = X € M. Thus, M is ao-algebra because the three conditions are satisfied. This means that (X, M) is a measurable space. Exercise 1. Use the same proving technique used in showing T = p(X) is a topology to show that M = p(X) is ao-algebra.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section: Chapter Questions
Problem 44RE
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Please answer the Exercise 1. Use the same proving technique used in showing ? = ℘(?) is a topology to show that ℳ = ℘(?) is a ?-algebra
88% i 3:03 PM Real Analysis Module 2 (in. LEARNING CONTENTS (The Concept of Measurability) Definition. (a) A collection M of subsets of a set X is said to be a o-algebra in X if M has the following properties: (i) X € M. (ii) If A E M, then AC € M, where AC is the complement of A relative to X. (i) If A = U A, and if A, E M for n = 1,2,3, ., then A € M. (b) If M is a o-algebra in X, then the ordered pair (X, M) is called a measurable space, and the members of M are called the measurable sets in X. (c) If X is a measurable space, Y is a topological space, and f is a mapping of X into Y, then f is said to be measurable provided that f-'(V) is a measurable seti X for every open set V in Y. PANGASINAN STATE UNIVERSITY Study Guide in Real Analysis FM-AA-CIA-15 Rev. 0 03-June-2020 Math Elective 2 (Real Analysis) Module 2: Abstract Integration Example. 1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = Ø is in M. Since M only contains ø and X then ØUX = X € M. Thus, M is ao-algebra because the three conditions are satisfied. This means that (X,M) is a measurable space. Exercise 1. Use the same proving technique used in showingT = p(X) is a topology to show that M = p(X) is a o-algebra. The most familiar topological spaces are the metric space. We shall assume some familiarity but shall give some basic definitions to recall this concept. Definition. A metric space (X,p) is a set X with a distance function or metric p satisfying the following nronerties Property (4) is called the triangle inequality.
Transcribed Image Text:88% i 3:03 PM Real Analysis Module 2 (in. LEARNING CONTENTS (The Concept of Measurability) Definition. (a) A collection M of subsets of a set X is said to be a o-algebra in X if M has the following properties: (i) X € M. (ii) If A E M, then AC € M, where AC is the complement of A relative to X. (i) If A = U A, and if A, E M for n = 1,2,3, ., then A € M. (b) If M is a o-algebra in X, then the ordered pair (X, M) is called a measurable space, and the members of M are called the measurable sets in X. (c) If X is a measurable space, Y is a topological space, and f is a mapping of X into Y, then f is said to be measurable provided that f-'(V) is a measurable seti X for every open set V in Y. PANGASINAN STATE UNIVERSITY Study Guide in Real Analysis FM-AA-CIA-15 Rev. 0 03-June-2020 Math Elective 2 (Real Analysis) Module 2: Abstract Integration Example. 1. Consider the collection M = {0, X}. Obviously, X E M. Note that X = Ø is in M. Since M only contains ø and X then ØUX = X € M. Thus, M is ao-algebra because the three conditions are satisfied. This means that (X,M) is a measurable space. Exercise 1. Use the same proving technique used in showingT = p(X) is a topology to show that M = p(X) is a o-algebra. The most familiar topological spaces are the metric space. We shall assume some familiarity but shall give some basic definitions to recall this concept. Definition. A metric space (X,p) is a set X with a distance function or metric p satisfying the following nronerties Property (4) is called the triangle inequality.
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