## Linear Operators: General theory |

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Page 100

Let fi be a positive extended real valued additive set function defined on a field E

of subsets of a set S. Then Proof. ... It follows immediately from Lemma 10 that

every subset of a ^a-

Let fi be a positive extended real valued additive set function defined on a field E

of subsets of a set S. Then Proof. ... It follows immediately from Lemma 10 that

every subset of a ^a-

**null set**, and every finite union of ^t-**null sets**, is a /4-**null set**.Page 147

A set is a

fi. F) = 0. Proof. If E is a

En containing E with v(/i, E„) < Thus the set F = PI E„ is a measurable set ...

A set is a

**null set**if and only if it is a subset of some measurable set F such that v(fi. F) = 0. Proof. If E is a

**null set**, then v*(ft. E) = 0 and there are measurable setsEn containing E with v(/i, E„) < Thus the set F = PI E„ is a measurable set ...

Page 213

(a) // for each p in a set AQG A(C) lim inf < r, MQ-*0 where C is a closed cube

containing p, then each neighborhood of A contains an open set Q such that A—

Q is a ft-

then ...

(a) // for each p in a set AQG A(C) lim inf < r, MQ-*0 where C is a closed cube

containing p, then each neighborhood of A contains an open set Q such that A—

Q is a ft-

**null set**and X(Q) < rfi(Q). (b) If for each p in a set AQG HC) hm sup > r,then ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero