## Linear Operators: General theory |

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Page 100

Q.E.D. Definitions 4 and 9 allow us to introduce one of the most frequently

occurring concepts in measure theory, namely, the concept of a

done in the following definition. 11 Definition. Let // be an additive set function

defined on ...

Q.E.D. Definitions 4 and 9 allow us to introduce one of the most frequently

occurring concepts in measure theory, namely, the concept of a

**null set**. This isdone in the following definition. 11 Definition. Let // be an additive set function

defined on ...

Page 147

A set is a

i, F) = 0. Proof. If E is a

containing E with v(ju, En) < 1/n. Thus the set F = D -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([i, F) = 0. Proof. If E is a

**null set**, then v*(ji, E) = 0 and there are measurable sets E„containing E with v(ju, En) < 1/n. Thus the set F = D -E„ is a measurable set ...

Page 213

(a) // for each p in a set AQG lim inf < r, Ai(C)-M) U(C) xohere C is a closed cube

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

Q is a u-

...

(a) // for each p in a set AQG lim inf < r, Ai(C)-M) U(C) xohere C is a closed cube

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

Q is a u-

**null set**and < r/u(Q). (b) // for each p in a set AQG .. A(C) hm sup — — > r,...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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### Common terms and phrases

Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero