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By Howard J. Wilcox

Undergraduate-level advent to Riemann essential, measurable units, measurable capabilities, Lebesgue essential, different themes. quite a few examples and workouts.

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8 If B C A. B measurable, f:A on B . 9 If f:A U B -. 6?. is measurable on on A U B, on A n B, and on A \B. _. 6?. 10 If f is measurable on A and m(B) = 0, show that f is measurable on A U B. 1 -+ 6l and f- (G) is measura­ G C 6l, then f is measurable on A . 1 1 Prove the remainder of Theorem 1 7. 1 2 Let C be the Cantor set (Example l 2. 2 1 ). Let D C [ 0, 1 ] be a nowhere dense measurable set with m(D) > 0 (Exercise 1 6. 22). 29). At each stage of the construction of C and of D, a certain finite number of open intervals of [ 0, 1 ] are deleted (put into [ 0, 1 ] \C or [ 0, 1 ] \D).

Proof: Open sets are countable pairwise disjoint unions of open intervals, which are measurable. Closed sets are complements of open sets. 8 Corollary: There exist non-measurable sets. 7 we produced pairwise disjoint sets - .. rr=l:1 m*(V,) would be equal outer measure. Since V, all of mU=1 V, = E, if the V, were measurable sets, 1 . This is clearly impossible, so the V, 's are not measurable. (V,) < m*( V,) for each of these sets. 0 A 1 C A 2 C A 3 C • • • are measurable subsets ofE, then U A;) = �im m*(A1).

E b;Xs,· , where the B; r= 1 are pairwise disjoint measurable sets with union A , then g is simple on A . 28 Show that every step function on false. 29 Give an example of a simple function and two different representations of it. 30 Prove Proposition 1 9. 3 . Given representations f = . E b;Xs ; and n r= 1 g = E k= l ck Xc k • find explicit representations of f + g, fg, f/g(g ::1= 0). 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.

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