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Page 164
... non - negative set function on to Σ is non - negative , it follows that { ū , ( E ) } is a bounded non - decreasing set of real numbers for each E e 21. We define 2 ( E ) lim , n ( E ) , E e 2. By Corollary 4 , 2 , is countably additive ...
... non - negative set function on to Σ is non - negative , it follows that { ū , ( E ) } is a bounded non - decreasing set of real numbers for each E e 21. We define 2 ( E ) lim , n ( E ) , E e 2. By Corollary 4 , 2 , is countably additive ...
Page 179
... non- negative u - measurable function defined on S and Let ¿ ( E ) = √2 f ( s ) u ( ds ) , E ΕΕΣ . g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and E √2g ( s ) 2 ( ds ) = √¿¡ ( s ) g ( s ) μ ...
... non- negative u - measurable function defined on S and Let ¿ ( E ) = √2 f ( s ) u ( ds ) , E ΕΕΣ . g be a non - negative 2 - measurable function defined on S. Then fg is u - measurable , and E √2g ( s ) 2 ( ds ) = √¿¡ ( s ) g ( s ) μ ...
Page 516
... non - negative measure μ defined for all Borel sets in S with the prop- erties that u is not identically zero and u is o - invariant . μ on S 40 Let S be a non - void set and G a family of functions to S. Suppose that 41 ( 42 ( 8 ) ...
... non - negative measure μ defined for all Borel sets in S with the prop- erties that u is not identically zero and u is o - invariant . μ on S 40 Let S be a non - void set and G a family of functions to S. Suppose that 41 ( 42 ( 8 ) ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ