## Linear Operators, Part 1 |

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Page 173

39

function

L1(S, 2, u). 40 Under the hypotheses of Exercise 36,

39

**Let**(S, X, u) be as in Exercise 38. Find a continuous monotone increasing realfunction

**f**defined on S such that**f**cannot be represented as**f**(s) = | g(t)at with g inL1(S, 2, u). 40 Under the hypotheses of Exercise 36,

**let**0 < x < 1, and suppose ...Page 223

Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes integral I =

so g(s)dh(s) exists.

c,d), with f(e)=a, f(d) = b. Show that the Lebesgue-Stieltjes integral | g(f(s)h(f(s) c ...

Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes integral I =

so g(s)dh(s) exists.

**Let f**be a continuous increasing function on an open interval (c,d), with f(e)=a, f(d) = b. Show that the Lebesgue-Stieltjes integral | g(f(s)h(f(s) c ...

Page 325

To prove (c),

F,). Let x, e F, and define f.(s) = x, for se E,. Then f. (modified if need be on a ...

To prove (c),

**let f**be u-measurable with u-essential bound B on E, and let e > 0.**Let F**. ..., F, be a covering of f(E) by disjoint Borel sets of scalars, and define E, -f-'(F,). Let x, e F, and define f.(s) = x, for se E,. Then f. (modified if need be on a ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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