1873_solutions

So 0 u pn f 0 l pn f 1 lim n 1 n2 0 lim n and

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Unformatted text preview: eltjes integrable with respect to  on a, b . 4. Suppose that x n is a sequence in an interval a, b and that x n has only finitely many partial limits. Suppose that  is an increasing function and that f is a bounded function on a, b that is continuous at every member of a, b that does not belong to the range of the sequence x n . Prove that f is Riemann-Stieltjes integrable with respect to  on a, b . Solution: We shall write the set of partial limits of junior Lebesgue criterion, suppose that  0. x n as y 1 , y 2 , , y k . To prove that f satisfies the For each j  1, 2, , k, we choose a number u j  y j and a number v j  y j such that  u j  J , y j  .  vj k We define 312 k U  uj, vj j1 and we observe that k k var , u j , v j var , U   j1 J , y j . j1 Since the set a, b U is closed and bounded and since the sequence x n has no partial limits in U. Therefore, if a, b U we know that x n cannot be frequently in the set a, b F  x n n  1, 2,  U then the set F is finite. We have thus found an elementary subset U Þ F of a, b such that var , U Þ F var , U  var , F k  J , y j   j1 and such that f is continuous at every number x a, b J , x xF UÞF . 5. This exercise does not ask you for a proof. Suppose that  is an increasing function, that x n is a sequence in an interval a, b and that f is a bounded function on a, b that is continuous at every member of a, b that does not belong to the range of the sequence x n . Do you think that the function f has to be integrable with respect to  on a, b . What does your intuition tell you? Solution: The function f must be integrable. This fact will follow from the full version of the Lebesgue criterion for integrability that will appear in the chapter on sets of measure zero. Some Exercises on the Composition Theorem 1. Given two functions f and g defined on a set S, we define the functions f f gx  fx if f x gx if f x gx if f x  g x g as follows: if f x  g x fx g and f gx and f gx  gx . Given that f and g are Riemann-Stieltjes integrable with respect to an increasing function  on an interval a, b , make the observations f  g  |f g | f g 2 and f  g |f g | f g 2 and deduce that the functions f g and f g are also integrable with respect to  on a, b . There really isn’t much to do in this exercise. The equation f x  g x  |f x g x | f gx 2 and f x  g x |f x g x | f gx 2 for each x follow at once when we consider the cases f x g x and f x  g x . 313 2. Given that f is a nonnegative function that is integrable on an interval a, b with respect to an increasing function , explain why the function f is integrable with respect to  on a, b . This exercise follows at once from the fact that the square root function is uniformly continuous on the range of f. 3. Given that f is integrable with respect to an increasing function  on an interval a, b , that f x 1 for every x a, b and that g x  log f x for every x a, b , explain why the function g must be integrable with respect to  on...
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