## Linear Operators: General theory |

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Page 126

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

fi 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

fi be ...

Page 132

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

X is also a finite,

follows ...

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

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

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

Page 136

{Hahn extension) Every

set function fx on a field E has a

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

{Hahn extension) Every

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

**countably additive**non-negative extension to thea-field determined by E. If fi 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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