## Linear Operators: General theory |

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Page 95

Finitely Additive Set

complex

be a ...

Finitely Additive Set

**Functions**( E ) = In contrast to the terms real**function**,complex

**function**, etc. , where the ... Thus , even if the integration process is**defined**with respect to a scalar valued set**function**u , the resulting integral v maybe a ...

Page 98

The total variation of an additive set

S is also additive on E. PROOF . Let { A ; } be a finite sequence of disjoint sets in

with A ÇEUF where E , F € ? and EF = $ . Let E ; = EA , F ; = FA ;, then \ u ( A ) ...

The total variation of an additive set

**function defined**on a field of subsets of a setS is also additive on E. PROOF . Let { A ; } be a finite sequence of disjoint sets in

with A ÇEUF where E , F € ? and EF = $ . Let E ; = EA , F ; = FA ;, then \ u ( A ) ...

Page 182

Q.E.D. Note : The unique u - integrable

the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da /

du . Thus da du is

Q.E.D. Note : The unique u - integrable

**function**f of Theorems 2 and 7 is calledthe Radon - Nikodým derivative of 2 with respect to u and is often denoted by da /

du . Thus da du is

**defined**u - almost everywhere by the formula ( d . à ( E ) ( s ) ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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