## Linear Operators, Part 1 |

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Page 177

Nelson Dunford, Jacob T. Schwartz. E C En may be

. A real valued set function can be represented as the difference of its positive

and negative variations ( 4.11 ) and so we may also

Nelson Dunford, Jacob T. Schwartz. E C En may be

**assumed**that 2 is real valued. A real valued set function can be represented as the difference of its positive

and negative variations ( 4.11 ) and so we may also

**assume**that 2 is positive .Page 178

Thus we may and shall

- measurable there is a sequence { gn } of simple functions converging to g ( s )

for every s in F except on a set E C F with v ( 2 , E ) = 0 ( by Corollary 6.13 ( a ) ) .

Thus we may and shall

**assume**that v ( u , F ) < oo and v ( a , F ) < 0 . Since g is a- measurable there is a sequence { gn } of simple functions converging to g ( s )

for every s in F except on a set E C F with v ( 2 , E ) = 0 ( by Corollary 6.13 ( a ) ) .

Page 535

Show that if f and g are non - negative functions in L ( S , E , u ) , then If + glo lilo +

glo . ( Hint . Use Exercise III.9.29 . ) 36 Let ( S , E , u ) and ( S1 , E1 , Mi ) be

positive measure spaces .

measurable ...

Show that if f and g are non - negative functions in L ( S , E , u ) , then If + glo lilo +

glo . ( Hint . Use Exercise III.9.29 . ) 36 Let ( S , E , u ) and ( S1 , E1 , Mi ) be

positive measure spaces .

**Assume**u ( S ) = 1. Show that if K is a u x My -measurable ...

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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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