## 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 cr-field of subsets of a set. In this case theresults of the preceding sections can be considerably extended. 1 Definition.

Page 132

This lemma shows that if each member of a generalized sequence {Xa} of finite,

X is also a finite,

This lemma shows that if each member of a generalized sequence {Xa} of finite,

**countably additive**measures is //-continuous and if ]\mx Xa(E) = X(E), EeE, whereX is also a finite,

**countably additive**measure, then X is also //-continuous.Page 136

(Hakn extension) Every

set function /t on a field E has a

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

(Hakn extension) Every

**countably additive**non-negative extended real valuedset function /t on a field E has a

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero