## Linear Operators, Part 1 |

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Page 186

Thus, the field (p of Lemma I is a g-field and the measure u of Lemma I is

countably additive on p. Consequently, the ... Let (S, X, u) be the product of finite

So the ...

Thus, the field (p of Lemma I is a g-field and the measure u of Lemma I is

countably additive on p. Consequently, the ... Let (S, X, u) be the product of finite

**positive measure spaces**(S1, 21, ul) and (S2, 22, us). For each E in 2 and so inSo the ...

Page 302

Let (S, 2, u) be a

Lp(S, 2, u), 1 < p < 00, is a complete lattice. PRoof. It is evidently sufficient to

show that if {f.} is a set of functions in Li such that 0 < f, s go for some goe L.1,

then sup ...

Let (S, 2, u) be a

**positive measure space**. Then the real partially ordered spaceLp(S, 2, u), 1 < p < 00, is a complete lattice. PRoof. It is evidently sufficient to

show that if {f.} is a set of functions in Li such that 0 < f, s go for some goe L.1,

then sup ...

Page 725

n—1 lim sup I. X. u(q-'e) < Ku(e) n—-co "joo for each set e of finite u-measure. (

Hint. Consider the map f(s) → X1(s)f(ps) for each A e 2 with u(A) < 00.) 37 Let (S.

2, u) be a

...

n—1 lim sup I. X. u(q-'e) < Ku(e) n—-co "joo for each set e of finite u-measure. (

Hint. Consider the map f(s) → X1(s)f(ps) for each A e 2 with u(A) < 00.) 37 Let (S.

2, u) be a

**positive measure space**, and T a non-negative linear transformation of...

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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