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3.3) Let X be an uncountable set and let A be the collection of subsets A of X such that either A or Aº is countable. Define µ(A) = 0 if A is countable and µ(A) = 1 if A is uncountable. Prove that u is a measure.

3.3) Let X be an uncountable set and let A be the collection of subsets A of X such that either A or Aº is countable. Define µ(A) = 0 if A is countable and µ(A) = 1 if A is uncountable. Prove that u is a measure.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Real Analyis question 3.3 dealing with measures. Please explain, thankyou in advance.

Measure: non-negative set function on a o - algebra A of subsets of some space X. Properties: 1) μ (φ) - 0 2) If A; E A and A¡ N A; = Ø; i j, for i, j E N = µ(U, A;) = E, µ(A;)...ountable additivity i=1 3.3) Let X be an uncountable set and let A be the collection of subsets A of X such that either A or A° is countable. Define µ(A) = 0 if A is countable and µ(A) = 1 if A is uncountable. Prove that u is a measure.
Transcribed Image Text:Measure: non-negative set function on a o - algebra A of subsets of some space X. Properties: 1) μ (φ) - 0 2) If A; E A and A¡ N A; = Ø; i j, for i, j E N = µ(U, A;) = E, µ(A;)...ountable additivity i=1 3.3) Let X be an uncountable set and let A be the collection of subsets A of X such that either A or A° is countable. Define µ(A) = 0 if A is countable and µ(A) = 1 if A is uncountable. Prove that u is a measure.
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