## Linear Operators: General theory |

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Page 112

A function f on S to X is

addition the equation lim ss \ 1m ( s ) –In ( s ) [ 0 \ u , ds ) = 0 . Such a sequence of

u ...

A function f on S to X is

**u**-**integrable**on S if there is a sequence { { n } of**u**-**integrable**simple functions converging to f in j - measure on S and satisfying inaddition the equation lim ss \ 1m ( s ) –In ( s ) [ 0 \ u , ds ) = 0 . Such a sequence of

u ...

Page 113

A function f on S is

integrable so is \ ( ( . ) ) . If { { n } is a sequence of - integrable simple functions

determining f in accordance with the preceding definition , then the sequence { \

ln ...

A function f on S is

**u**-**integrable**if and only if it is v (**u**) -**integrable**. If f is j -integrable so is \ ( ( . ) ) . If { { n } is a sequence of - integrable simple functions

determining f in accordance with the preceding definition , then the sequence { \

ln ...

Page 180

Sleds ) = f ( s ) g ( s ) u ( ds ) . Εε Σ . PROOF . Suppose that g is 2 - integrable . By

Lemma 3 , fg is measurable . The

when it is observed ( by Theorem 4 ) that \ | ( • ) g ( ) is

...

Sleds ) = f ( s ) g ( s ) u ( ds ) . Εε Σ . PROOF . Suppose that g is 2 - integrable . By

Lemma 3 , fg is measurable . The

**u**-**integrability**of tg follows from Theorem 2.22when it is observed ( by Theorem 4 ) that \ | ( • ) g ( ) is

**u**-**integrable**. Theorem 4...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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