## Linear Operators: General theory |

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Page 186

measurable. The function p1(E(s2)) is p2-integrable and 5 Corollary. The product

of finite

fact follows immediately from the formula for fi(E) given in Corollary 4. Q.E.D. It is

...

measurable. The function p1(E(s2)) is p2-integrable and 5 Corollary. The product

of finite

**positive measure spaces**is a finite**positive measure space**. Proof. Thisfact follows immediately from the formula for fi(E) given in Corollary 4. Q.E.D. It is

...

Page 530

for functions with values in the B-space Lp. In more analytical terms, we obtain

the following inequalities: 18 Let (SvE1,fi1) and (S2, E2. fJ2) be

for functions with values in the B-space Lp. In more analytical terms, we obtain

the following inequalities: 18 Let (SvE1,fi1) and (S2, E2. fJ2) be

**positive measure****spaces**, and let A' be a fix x ^-integrable function on Sl xS2. Then, for p 2: 1. f (f ...Page 725

I n-1 lim sup - 2 iM(99_'e) Knie) fl-*00 11 ;=0 for each set e of finite ^-measure. (

Hint. Consider the map f(s) -> xA(s)f((ps) for each ^4 e27 with < oo.) 87 Let (5, 2",

// ) be a

I n-1 lim sup - 2 iM(99_'e) Knie) fl-*00 11 ;=0 for each set e of finite ^-measure. (

Hint. Consider the map f(s) -> xA(s)f((ps) for each ^4 e27 with < oo.) 87 Let (5, 2",

// ) be a

**positive measure space**, and T a non-negative linear transformation of ...### 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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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact