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Problem 2. Let N denote the non-measurable subset of [0, 1], constructed in class and in...

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Advance Math

Problem 2. Let N denote the non-measurablesubset of [0, 1], constructed in class and in the book "RealAnalysis: Measure Theory, Integration, and Hilbert Spaces" by E. M.Stein, R. Shakarchi.

(a) Prove that if E is a measurable subset of N , then m(E) =0.

(b) Assume that G is a subset of R with m?(G) > 0,prove that there is a subset of G such that it isnon-measurable.

(c) Prove that if Nc = [0, 1] N , thenm?(Nc) = 1.

(d) Now, conclude that

m?(N ) + m?(Nc ) ? m?(N ?Nc ).

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