## 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 fi 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 fi 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 /< be a vector valued, complex valued, or

additive set function defined on a field E of subsets of a set S. Then ji is said to be

...

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

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

**extended real valued**additive set function defined on a field E of subsets of a set S. Then ji 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 compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions 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 Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact