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By Howard J. Wilcox

Undergraduate-level advent to Riemann fundamental, measurable units, measurable capabilities, Lebesgue quintessential, different subject matters. a number of examples and routines.

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Let e > 0, and let G be open in [c,d] with ( [c,d] \A ) C G and m*(G) < m *( [c,d] \A) + e. Then H = G U [a,c) is open in [a,b] , and ( [a,b] \A) C H, so m*( [a,b ] \A) < m *(G U [a,c)) = m*(G) + (c - a) < m*( [c,d] \A) + (c - a) + e. Since e was arbitrary, m*([a,b]\A) < m*([c,d]\A) + (c - a). On the other hand, if G ' is open in [a,b] , G ' ::) ( [a,b] \A), and m*(G ' ) < m * ([a,b] \A) + e, then [a,c) C G ', H ' = G '\ [a,c) is open in [c,d] , contains [c,d) \A , and m*([c,d]\A ) < m*(H ' ) = m*(G ') - (c - a) < m*([a,b]\A) - (c - a) + e.

IR are simple, so are f + g, fg, and ffg 0 Simplicity is not preserved under limits. 6. The converse i s also true. 4 Theorem : A function {:A -+ IR is measurable if and only iff is the point· wise limit of a sequence of simple functions on A . Proof: (•) . 6. (�) . Let Pn be a partition of [-n,n] (on the y -axis) ob· tained by taking equal sub-intervals of length 1/n. ) 2 n2 Now let B; = { x E A IYi - 1 < f(x) < y,} , and let In = � Yt - 1 "XB i · i- 1 where Yt = - n • Simp • 1e. Then 1I"n IS Now given x0 E A , for each n big enough so that f(x0) E [-n,n] , Yt- 1 < f(x o) < Yi for some i = 1 ,2, • • • ,2n 2 • Thus x0 E Bi and 1/(x o) - /n(X o) l = lf(x o ) - Yt- 1 1 < lY; -Yt- 1 1 = 1/n.

N A; is measurable and m ( nt A1) = limm*(A;). 10 Corollary : If - * - . 7. 1 1 Corollary : Let be any (not necessarily disjoint) countable col­ lection of measurable sets inE. Then - ,� A; is measurable and (2) ntAt is measurable. \( �Ut A ,] }U • • • . } . 6}. t [E\A;] ) and use part {1}. - - , 0 11. Summary Let us pause to take account of our accomplishments. Our original (ambitious) goal was to construct a set function defined for aU subsets of E 0,1 and satisfying m =[ ] (1} O < m(A) < 1 for any A C E.

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