1873_solutions

0 s n d n j1 1 2n 1n 2 a kj 3j n 1 2n n n n a

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Unformatted text preview: ry number  0 there exist functions g and h that are Riemann-Stieltjes integrable with respect to  on the interval a, b such that g f h and Þa h b g d  . To show that condition a implies condition b we assume that condition a holds. In other words, we assume that f is integrable with respect to  on a, b . Suppose that  0. Using the first criterion for integrability we choose a partition P of a, b such that Þ a w P, f d  b . We define S  u P, f and s  l P, f and observe that the functions s and S have the desired properties. The proof that condition a implies condition c is very similar. This time, the partition P is chosen using the second criterion for integrability. To prove that condition b implies condition a we assume that condition b holds. What we shall show is that the first criterion for integrability holds. Suppose that  0. Using condition b we choose step functions s and S on the interval a, b such that s f S and Þa S b s d  . Choose a partition P of a, b such that both s and S step within P and observe that since 308 s l P, f u P, f S we have Þ a w P, f d  Þ a u P, f b b l P, f d Þa S b s d  . The proof that condition c implies condition a is very similar. This time we show that f is integrable by showing that the second criterion for integrability holds. 4. Suppose that  is an increasing function, that f is a bounded function on an interval a, b and that for every number  0 there exists an elementary subset E of a, b such that var , E  and such that the function f 1  E is integrable with respect to  on a, b . Prove that f must be integrable with respect to  on the interval a, b . Solution: In order to show that f is integrable on a, b we shall show that f satisfies the second criterion for integrability. Suppose that  0. Using the given property of f we choose an elementary subset A of a, b such that var , A  /2 and such that the function f 1  A is integrable with respect to  on a, b . Choose a partition P 1 of a, b such that the function  A steps within P. Now, using the fact that f 1  A satisfies the second criterion for integrability we choose a partition P 2 of a, b such that if we definè B x a, b w P, f 1  A x . then var , B  /2. We now define P to be the common refinement of P 1 and P 2 and we express P as P  x 0 , x 1 , , x n . For each j  1, 2, , n, if the open interval x j 1 , x j is not included in A Þ B then, since the functions f and f 1  A agree in the interval x j 1 , x j the condition w P, f x  w P, f 1  A x  x j 1 , x j . Since var , A Þ B var , A  var , B    2 2 we have succeeded in showing that f satisfies the second criterion for integrability. must hold whenever x 5. a. Suppose that  is an increasing function, that f is a nonnegative function defined on an interval a, b and that for every number  0, the set x a, b fx is finite. Prove that f must be integrable with respect to  on a, b and that Þ fd  0. a We shall show that f satisfies the first criter...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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