By Howard J. Wilcox

Undergraduate-level advent to Riemann quintessential, measurable units, measurable capabilities, Lebesgue imperative, different issues. quite a few examples and routines.

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**Example text**

G(x)} is 58 L E B ESG U E I NT E G R A T I O N A N D F O U R I E R SE R I ES Show that h is measurable on A . 25 Let /,. be measurable on bounded measurable set A , for n = 1 ,2, . . (x)} converges} show that B is measurable. 26 Let /,. be measurable on A for n = 1 ,2, . . Let /,. on A . e. 1 :> c}) = 0. 27 Prove the remainder of Proposition 1 9. 2 ; if g = . 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.

1 2 Find m *(G), where G = (0, 1 ]\{ 1 /n j n = 1 , 2 , . . }. 13 Do there exist open subsets G � o G 2 of E such that G 1 :I= G2 but m *( G 1 ) = m *( G 2 )? 15 Prove that m * is countably additive on the class of open subsets of E. 4 ; that is, if [ a,b ] C E, show that I b - a = glb { m *(G> £a,b ] C G and G open in E}. (b) Is there an open set G C E with [ 1 /2,3/4] C G and m*(G) = t? 9. 1 7 Show that if A C B C E, then m *(A ) < m *(B). 1 8 If A C [ O,t J t l] and B C < . show that m *(A U B) = m *(A ) + m *(B).

Then there is m(G) m*(X) + e . 4] ) = m(G) < m*(X) + e since m is additive M disjoint measurable subsets of E. Since this inequality holds for every e > O,m*(X n A)+m*(X n [E\A. ] ) < m*(X). 0 The converse follows by letting X =E. One advantage of the Caratheodory Criterion is that it does not require the concept of inner measure. Thus it can be used to defme measure and measurability for subsets of tR . Recall that in Lemma 1 0. 1 we proved that if G1 and G2 are open in E, then m*(G 1) + m*(G2) > m*(G 1 U G2) + m*(G 1 n G2).