## Linear Operators, Part 1 |

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Page 3

that is , for every a € A , the

g : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A. If f : A → B and C CA , the symbol f ( C ) is used for the set of all ...

that is , for every a € A , the

**function f**assigns an element f ( a ) € B. If f : A + B andg : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A. If f : A → B and C CA , the symbol f ( C ) is used for the set of all ...

Page 196

For each s in S , F ( s ) is an equivalence class of functions , any pair of whose

members coincide 2 - almost everywhere . If for each s we select a particular

S ...

For each s in S , F ( s ) is an equivalence class of functions , any pair of whose

members coincide 2 - almost everywhere . If for each s we select a particular

**function f**( s , :) € F ( s ) , the resulting**function f**( s , t ) defined on ( R , ER , Q ) = (S ...

Page 199

integral Ss ( s , t ) u ( ds ) , as a

ds ) of L , ( T , ET , 2 , X ) . Proof . Let E , be partitioned into a sequence { En } of

disjoint sets of finite 2 - measure . For 1 spsoo let L , = L , ( T , ET , 2 , X ) and

define ...

integral Ss ( s , t ) u ( ds ) , as a

**function of**t , is equal to the element Ss F ( s ) q (ds ) of L , ( T , ET , 2 , X ) . Proof . Let E , be partitioned into a sequence { En } of

disjoint sets of finite 2 - measure . For 1 spsoo let L , = L , ( T , ET , 2 , X ) and

define ...

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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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