## Linear Operators, Part 1 |

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Page 95

Thus, even if the integration process is defined with respect to a scalar valued set

function u, the resulting integral v may be a

desirable therefore to formulate some of the basic and elementary concepts in

such a ...

Thus, even if the integration process is defined with respect to a scalar valued set

function u, the resulting integral v may be a

**vector valued**set function. It isdesirable therefore to formulate some of the basic and elementary concepts in

such a ...

Page 318

last section permit us to develop a more satisfactory theory of

countably additive set functions (briefly,

able to ...

**Vector Valued**Measures The theorems on spaces of set functions proved in thelast section permit us to develop a more satisfactory theory of

**vector valued**countably additive set functions (briefly,

**vector valued**measures) than we wereable to ...

Page 319

By Corollary III.7.4, the set {you} of scalar valued measures is uniformly countably

additive on 21, which contradicts the ... Q.E.D. V In contrast to the case of

complex valued measures, the total variation of a

Definition ...

By Corollary III.7.4, the set {you} of scalar valued measures is uniformly countably

additive on 21, which contradicts the ... Q.E.D. V In contrast to the case of

complex valued measures, the total variation of a

**vector valued**measure (cf.Definition ...

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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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### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero