1873_solutions

H f x 0 f x dx 1 1 dx h this inequality is known

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Unformatted text preview: Þ . Ý a 8. Given that f is a step function and that  and  are increasing functions, prove that Ý Þ Ý fd    Ý Ý Þ Ý fd  Þ Ý fd.  We begin by choosing an interval a, b and a partition P  x 0 , x 1 , , x n of a, b such that f x  0 whenever a number x lies outside the interval a, b and such that f steps within the partition P. If the constant value of f in each interval x j 1 , x j is  j then Ý Þ Ý fd    n n f x j J   , x j   j0    xj j n   1  n f xj j0 Ý    xj j1 J  , x j  J , x j  j  xj   xj  xj 1   xj 1  j1 Ý Þ Ý fd  Þ Ý fd. 9. Given that f is a step function and that  is an increasing function and that c is a nonnegative number, prove that Ý Þ Ý fd c  cÞ Ý Ý fd. We begin by choosing an interval a, b and a partition P  x 0 , x 1 , , x n of a, b such that f x  0 whenever a number x lies outside the interval a, b and such that f steps within the partition P. If the constant value of f in each interval x j 1 , x j is  j then 290 Ý Þ Ý fd c n n f x j J c, x j   j0 n c x j n f x j J , x j  c c c x j j 1  j1 j0 j  xj  xj 1  j1  cÞ Ý Ý fd. Exercises on Elementary Sets 1. Given that A and B are elementary sets and  is an increasing function, prove that var , A Þ B  var , A  var , B var , A B . Solution: We begin by choosing a lower bound a and an upper bound b of the set A Þ B. By looking at the different cases we can see easily that whenever x a, b we have  A ÞB   A   B  A B . Therefore var , A Þ B   Þ a  AÞB d  Þ a  A   B b b A B d Þ a  A d  Þ a  B d Þ a  A B d b b  var , A  var , B b var , A B. 2. Prove that if E is an elementary set and m E  0 then E must be finite. Choose an interval a, b such that the function  E is zero outside a, b and a partition P  x 0 , x 1 , , x n of a, b such that  E steps within P. Since  E is nonnegative and its sum over P is zero, the constant value of  E in each interval x j 1 x j must be zero. Therefore  E can be nonzero only at points of P, in other words, every member of E is a point of P and we conclude that the set E is finite. 3. Explain why the set of all rational numbers in the interval 0, 1 is not elementary. The desired result will be clear when we have proved the stronger assertion that is made in Exercise 4. 4. Prove that if E is an elementary subset of 0, 1 and if every rational number in the interval belongs to E then the set 0, 1 E must be finite. We assume that E is an elementary subset of 0, 1 and that every rational number in 0, 1 belongs to E. Choose a partition P  x 0 , x 1 , , x n of a, b such that  E steps within P. For each j, since the interval x j 1 , x j contains some rational numbers, the constant value of  E in x j 1 , x j must be 1. Since the numbers in 0, 1 that do not belong to E have to be points of P, the set 0, 1 E must be finite. 5. Give an example of a set A of numbers such that if E is any elementary subset of A we have...
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