## Linear Operators: General theory |

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Page 96

A

if = 0 and /<Miu4 • -lMJ =/«(^1)+/u(^2)+. . •+1«(^„), for every finite family {^j, . . ., An

} of disjoint subsets of r whose union is in t. For an example of a finitely

A

**set function**defined on a family t of**sets**is said to be**additive**or finitely**additive**if = 0 and /<Miu4 • -lMJ =/«(^1)+/u(^2)+. . •+1«(^„), for every finite family {^j, . . ., An

} of disjoint subsets of r whose union is in t. For an example of a finitely

**additive**...Page 97

The total variation v(ii) of an

dominates fx in the sense that v(/n, E) Ji j/u(£)| for E e E; the reader should test his

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

The total variation v(ii) of an

**additive set function**/n is important because itdominates fx in the sense that v(/n, E) Ji j/u(£)| for E e E; the reader should test his

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

Page 126

Countably

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

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.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

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