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Finite or countable collection of disjoint open intervals and Cantor sets.

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a) If {I } is a finite or countable collection of disjoint open intervals with
I (a, b), Prove that (m: Measure).

b) If U (empty set) is an open set of R, Prove That (U)> 0.

c) Let P denote the cantor set in [0, 1]. Prove that m(P) = 0.

d) Suppose that U, V are open subsets of R, a,b R with U [a,b] = V [a,b]. Prove that m(U [a,b]) = m(V [a,b]).

e) If A,B are subsets of R, prove that = , = , and (

( ( of complement of A) and is Characteristic function of a set).

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Finite or countable collection of disjoint open intervals and Cantor sets are investigated. The solution is detailed and well presented.

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