Linear Operators: General theory |
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Page 106
... measurable , the function f is said to be μ - measurable or , if u is understood , simply measurable . A set A is measurable if XA is measurable . Symbols for the set of measurable functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ...
... measurable , the function f is said to be μ - measurable or , if u is understood , simply measurable . A set A is measurable if XA is measurable . Symbols for the set of measurable functions are M ( S , Σ , μ , X ) , M ( S , Σ , μ ) , M ...
Page 179
... measurable function defined on S and ¿ ( E ) = √ ̧ f ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is μ - measurable , and √gg ( s ) λ ( ds ) = √ ¿ f ( s ) g ( s ) μ ( ds ) , ΕΕΣ ...
... measurable function defined on S and ¿ ( E ) = √ ̧ f ( s ) μ ( ds ) , E ΕΕΣ . Let g be a non - negative 2 - measurable function defined on S. Then fg is μ - measurable , and √gg ( s ) λ ( ds ) = √ ¿ f ( s ) g ( s ) μ ( ds ) , ΕΕΣ ...
Page 180
... measurable function is the difference of two non- negative measurable functions , it follows from the theorem that [ * ] √2g ( s ) λ ( ds ) = √ £ f ( s ) g ( s ) μ ( ds ) , ΕΕΣ , for a real measurable function g . Since the real and ...
... measurable function is the difference of two non- negative measurable functions , it follows from the theorem that [ * ] √2g ( s ) λ ( ds ) = √ £ f ( s ) g ( s ) μ ( ds ) , ΕΕΣ , for a real measurable function g . Since the real and ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ