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-null set, and every finite union of ^t-null sets, is a /4-null set.
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 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 sets
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 sets
En 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- null set and X(Q) < rfi(Q). (b) If for each p in a set AQG HC) hm sup > r,
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 | |
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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
Bibliographic information
| Title | Linear Operators: General theory Volume 7 of Pure and applied mathematics Volume 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics Interscience Press |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Length | 2592 pages |
| Subjects | › › Mathematics / Algebra / Linear Mathematics / Calculus |
| Export Citation | BiBTeX EndNote RefMan |