## Linear Operators, Part 1 |

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Page 126

results of the preceding sections can be considerably extended . DEFINITION .

**Countably Additive**Set Functions The basis for the present section is a**countably****additive**set function defined on a o - field of subsets of a set . In this case theresults of the preceding sections can be considerably extended . DEFINITION .

Page 132

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

where 2 is also a finite ,

...

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

**countably additive**measures is u - continuous and if lim , 22 ( E ) = 2 ( E ) , E € ,where 2 is also a finite ,

**countably additive**measure , then î is also u - continuous...

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( Hahn extension ) Every

set function u on a field E has a

the o - field determined by E. If u is o - finite on { then this extension is unique .

( Hahn extension ) Every

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

**countably additive**non - negative extension tothe o - field determined by E. If u is o - finite on { then this extension is unique .

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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