Linear Operators: General theory |
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Page 164
... non - negative set function on to 2 is non - negative , it follows that { ( E ) } is a bounded non - decreasing set of real numbers for each E € 21. We define ¿ 1 ( E ) lim , n ( E ) , E € 21. By Corollary 4 , 21 is countably additive ...
... non - negative set function on to 2 is non - negative , it follows that { ( E ) } is a bounded non - decreasing set of real numbers for each E € 21. We define ¿ 1 ( E ) lim , n ( E ) , E € 21. By Corollary 4 , 21 is countably additive ...
Page 179
... non- negative u - measurable function defined on S and λ ( E ) = √ f ( s ) μ ( ds ) , ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and = √pg ( s ) λ ( ds ) Spf ( s ) g ( s ) μ ( ds ) ...
... non- negative u - measurable function defined on S and λ ( E ) = √ f ( s ) μ ( ds ) , ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and = √pg ( s ) λ ( ds ) Spf ( s ) g ( s ) μ ( ds ) ...
Page 516
... non - negative measure μ defined for all Borel sets in S with the prop- erties that is not identically zero and u is o - invariant . 40 Let S be a non - void set and G a family of functions on S to S. Suppose that 41 ( 42 ( 8 ) ) = 42 ...
... non - negative measure μ defined for all Borel sets in S with the prop- erties that is not identically zero and u is o - invariant . 40 Let S be a non - void set and G a family of functions on S to S. Suppose that 41 ( 42 ( 8 ) ) = 42 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ