## Linear Operators: General theory |

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Page 95

A discussion of some of the deeper properties of vector

contained in Section IV. 10. It is desirable also to allow the set function ft to have

its values in the

A discussion of some of the deeper properties of vector

**valued**set functions iscontained in Section IV. 10. It is desirable also to allow the set function ft to have

its values in the

**extended real**number system (which is not a vector space) but, ...Page 126

In this case the results of the preceding sections can be considerably extended. 1

Definition. Let /t be a vector valued, complex valued, or

additive set function defined on a field 27 of subsets of a set S. Then p, is said to

be ...

In this case the results of the preceding sections can be considerably extended. 1

Definition. Let /t be a vector valued, complex valued, or

**extended real valued**additive set function defined on a field 27 of subsets of a set S. Then p, is said to

be ...

Page 137

12 Lemma. The total variation of a regular additive complex or

variations of a bounded regular real valued additive set function are also regular.

Proof.

12 Lemma. The total variation of a regular additive complex or

**extended real****valued**set function on a field is regular. Moreover, the positive and negativevariations of a bounded regular real valued additive set function are also regular.

Proof.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero