Linear Operators: General theory |
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Page 164
... non - negative set function on to 1 is non - negative , it follows that { ( E ) } is a bounded non - decreasing set of real numbers for each E e 21. We define ( E ) lim , μn ( E ) , E € 21. By Corollary 4 , 21 is countably additive on ...
... non - negative set function on to 1 is non - negative , it follows that { ( E ) } is a bounded non - decreasing set of real numbers for each E e 21. We define ( E ) lim , μn ( E ) , E € 21. By Corollary 4 , 21 is countably additive on ...
Page 179
... non- negative u - measurable function defined on S and ¿ ( E ) = √2 † ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative λ - measurable function defined on S. Then fg is u - measurable , and - g ( s ) 2 ( ds ) = f ( s ) g ( s ) μ ( ds ) ...
... non- negative u - measurable function defined on S and ¿ ( E ) = √2 † ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative λ - measurable function defined on S. Then fg is u - measurable , and - g ( s ) 2 ( ds ) = f ( s ) g ( s ) μ ( ds ) ...
Page 516
... non - negative measure u defined for all Borel sets in S with the prop- erties that μ is not identically zero and u is o - invariant . μ 40 Let S be a non - void set and G a family of functions & on S to S. Suppose that 41 ( 2 ( 8 ) ...
... non - negative measure u defined for all Borel sets in S with the prop- erties that μ is not identically zero and u is o - invariant . μ 40 Let S be a non - void set and G a family of functions & on S to S. Suppose that 41 ( 2 ( 8 ) ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ