## Linear Operators: General theory |

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Page 126

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.

Page 136

(Hahn extension) Every

set function ft on a field E lias a

a-field determined by E. If ft is a-finite on E then this extension is unique. Proof.

(Hahn extension) Every

**countably additive**non-negative extended real valuedset function ft on a field E lias a

**countably additive**non-negative extension to tliea-field determined by E. If ft is a-finite on E then this extension is unique. Proof.

Page 405

The measure it^ is

measure li is not

is easy to see that l2 is the union of a countable family of yu-null sets.

The measure it^ is

**countably additive**; the subset ^ of * is of /^-measure zero; themeasure li is not

**countably additive**. In fact, using the relation between fi^ and ii, itis easy to see that l2 is the union of a countable family of yu-null sets.

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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