## Linear Operators, Part 1 |

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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 g-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 {2,} of finite,

2 is also a finite,

This lemma shows that if each member of a generalized sequence {2,} of finite,

**countably additive**measures is u-continuous and if lim, 2.(E) = Z(E), Ee X, where2 is also a finite,

**countably additive**measure, then A is also u-continuous.Page 136

(Hahn extension) Every

set function u on a field 2 has a

g-field determined by 2. If u is o-finite on 2 then this eartension is unique. PROOF.

(Hahn extension) Every

**countably additive**non-negative eatended real valuedset function u on a field 2 has a

**countably additive**non-negative extension to theg-field determined by 2. If u is o-finite on 2 then this eartension is unique. PROOF.

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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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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