## Linear Operators: General theory |

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Page 96

A

family {Av . . ., An) of disjoint subsets of r whose union is in t. For an example of a

finitely ...

A

**set function**(i defined on a family z of**sets**is said to be**additive**or finitely**additive**if u(tf>) = 0 and u(Ax\jAv ..\jAn) = fi{A1)+/j,(Ai)+. . .+u(An), for every finitefamily {Av . . ., An) of disjoint subsets of r whose union is in t. For an example of a

finitely ...

Page 97

The total variation v(p) of an

dominates ft in the sense that v(fi, E) 2r for E e E; the reader should test his

comprehension of Definition 4 below by proving that v(ft) is the smallest of the

non-negative ...

The total variation v(p) of an

**additive set function**fi is important because itdominates ft in the sense that v(fi, E) 2r for E e E; the reader should test his

comprehension of Definition 4 below by proving that v(ft) is the smallest of the

non-negative ...

Page 126

Countably

results of the preceding sections can be considerably extended. 1 Definition. Let

u be ...

Countably

**Additive Set Functions**The basis for the present section is a countably**additive set function**defined on a a-field of subsets of a set. In this case theresults of the preceding sections can be considerably extended. 1 Definition. Let

u be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero