By Howard J. Wilcox

Undergraduate-level advent to Riemann critical, measurable units, measurable features, Lebesgue crucial, different themes. a variety of examples and workouts.

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Undergraduate-level advent to Riemann imperative, measurable units, measurable capabilities, Lebesgue quintessential, different subject matters. various examples and workouts.

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31 Suppose g is simple and f is any function. Under what conditions is f • g simple'? When is g · { simple'? 32 Prove that if f is measurable on A and f is bounded below, then / is the pointwise limit of an increasing sequence of simple functions. (Hint : at stage n , partition [ -n, n ] into 2n • 2" equal subintervals. 33 Find an increasing sequence of simple functions on [ 0, 1 ] with limit /(x) = x. 34 (a) Show that the simple function Xc (where C is the Cantor set) is the pointwise limit of a decreasing sequence of step functions, but not of an in creasing sequence of step functions.

7 Prove Corollary 1 0. 1 0. 8 If m(A ) = 0 and {:E -+ E has bounded derivative, show that m(f(A )) = 0. (Hint : use the mean value theorem on each open interval of an open set G containing A . ) S C E, then there is a measurable set A C E such that S C A and m (A ) = m*(S). 10 Prove Proposition 1 2. 2. 1 1 Verify that the following collections are countably additive classes of sets: (a) (b) the collection of all subsets of E. the collection of all measurable subsets of E. 1 2. 1 is called "closure under countable unions.

A 1 n A2). Proof : Given e > 0, there exist open sets G1, G 2 C E such that G1 :> A 1, G 2 :> A 2 , and m*(Gi) < m*(A1) + e for i = 1 ,2. Thus <. s 0. <. m*(A 1) + e + m*(A 2 ) + e - m*(A 1 n A 2 ). Since this is true for all e > 0, the result in (1) follows. 2). 2 Lemma: If A 1 and A 2 are measurable subsets ofE, then A 1 U A 2 and A 1 n A 2 are measurable. (A 2 ). (A1) by hypothesis, so the inequalities become equal signs. (A 1 n A 2 ). (A 2). Proof: Since m*(�) = 0, this is obvious from the Lemm a..