Linear Operators: General theory |
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Page 100
... null set . This is done in the following definition . = 11 DEFINITION . Let μ be an additive set function defined on a field of subsets of a set S. A subset N of S is said to be a u - null set if v * ( μ , N ) 0 , where * is the ...
... null set . This is done in the following definition . = 11 DEFINITION . Let μ be an additive set function defined on a field of subsets of a set S. A subset N of S is said to be a u - null set if v * ( μ , N ) 0 , where * is the ...
Page 147
... set is a null set if and only if it is a subset of some measurable set F such that v ( μ , F ) 0 . PROOF . If E is a null set , then v * ( μ , E ) O and there are mea- surable sets En containing E with v ( μ , En ) < 1 / n . Thus the set ...
... set is a null set if and only if it is a subset of some measurable set F such that v ( μ , F ) 0 . PROOF . If E is a null set , then v * ( μ , E ) O and there are mea- surable sets En containing E with v ( μ , En ) < 1 / n . Thus the set ...
Page 213
... null set and λ ( Q ) < rμ ( Q ) . ( b ) If for each p in a set ACG λ ( C ) > r , lim sup μ ( C ) → 0 μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a u - null set and λ ( B ) > ru ( B ) . PROOF . To prove ...
... null set and λ ( Q ) < rμ ( Q ) . ( b ) If for each p in a set ACG λ ( C ) > r , lim sup μ ( C ) → 0 μ ( C ) then each neighborhood of A contains a Borel set B such that A - B is a u - null set and λ ( B ) > ru ( B ) . PROOF . To prove ...
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