## Linear Operators: General theory |

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Page 126

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

t be ...

**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. Let /

t be ...

Page 132

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

where A is also a finite,

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

**countably additive**measures is ^-continuous and if lim,,, Xa(E) = X(E), EeZ,where A is also a finite,

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

(Hahn extension) Every

set function pi on a field E has a

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

(Hahn extension) Every

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

**countably additive**non-negative extension to thea-field determined by E. If p. 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 | 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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero