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

Problem 2. Let N denote the non-measurable subset of [0, 1], constructed in class and in the book "Real Analysis: 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 is non-measurable.

(c) Prove that if N^c = [0, 1] \ N , then m_∗(N^c) = 1.

(d) Now, conclude that

m_∗(N ) + m∗(N^c ) ≠ m_∗(N ∪ N^c ).

Expert Solution

Colby Messinger answered 1 year ago

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