## Linear Operators: General theory |

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12

T, ZT, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e E

) is a fi-null set. Proof. By

12

**Lemma**. Let (R, ZR, q) be the product of f inite measure spaces (S, Z, yu) and (T, ZT, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e E

) is a fi-null set. Proof. By

**Lemma**11 it may be assumed that the measure ...Page 699

1 f° _ 2j_ , 1 e~v do = - \/n 2 Therefore Vi □ = ar»/*<; \*y/m VM) fa 2y2' 2V^Jo ^/SI

2-v/: 2<r-V« /•<» ^ _/Vl*_\« — e V«« I dy Jo 2y9 V71 Jo A = «-V«, which proves (*)

and completes the proof of the

1 f° _ 2j_ , 1 e~v do = - \/n 2 Therefore Vi □ = ar»/*<; \*y/m VM) fa 2y2' 2V^Jo ^/SI

2-v/: 2<r-V« /•<» ^ _/Vl*_\« — e V«« I dy Jo 2y9 V71 Jo A = «-V«, which proves (*)

and completes the proof of the

**lemma**. Q.E.D. We shall now state and prove the ...Page 700

amJ0 Jo a* Jo Jo This shows that if the

it is also true for any integer m ^ ft. Thus, to prove the

that it is true if k is even. If ft = 1 the

amJ0 Jo a* Jo Jo This shows that if the

**lemma**is known to be true for an integer ftit is also true for any integer m ^ ft. Thus, to prove the

**lemma**, it will suffice to showthat it is true if k is even. If ft = 1 the

**lemma**has already been proved (**Lemma**...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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