## Linear Operators, Part 1 |

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Page 193

12

Xr, Ž). Let E be a g-null set in R. Then for Ž-almost all t, the set E(t) = {s}ss, t e E} is

a u-null set. PRoof. By

12

**LEMMA**. Let (R, 2F, 0) be the product of finite measure spaces (S, X, u) and (T,Xr, Ž). Let E be a g-null set in R. Then for Ž-almost all t, the set E(t) = {s}ss, t e E} is

a u-null set. PRoof. By

**Lemma**11 it may be assumed that the measure spaces ...Page 697

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. 11

be a strongly measurable semi-group of operators in L1(S. 2, u) with T(t1,..., to)|1

< 1, ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. 11

**LEMMA**. Let (S. 2, u) be a positive measure space and let {T(t1,..., ti), ti, ..., to > 0}be a strongly measurable semi-group of operators in L1(S. 2, u) with T(t1,..., to)|1

< 1, ...

Page 699

dy Vot Jo 2y” 2e-vo - - A - eTV t, Vot which proves (*) and completes the proof of

the

technical reasons occurring later the following

...

dy Vot Jo 2y” 2e-vo - - A - eTV t, Vot which proves (*) and completes the proof of

the

**lemma**. Q.E.D. We shall now state and prove the**lemma**referred to as CP. Fortechnical reasons occurring later the following

**lemma**is stated for what might be...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero