Linear Operators: General theory |
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Page 108
... u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation [ _h ( s ) u ( ds ) = √ f ( s ) u ( ds ) S E E = n Σπ.μ ( ΕΕ ; ) . i = 1 ...
... u - integrable simple function " will be used interchan- geably . For an Ee the integral over E of a u - integrable simple func- tion h is defined by the equation [ _h ( s ) u ( ds ) = √ f ( s ) u ( ds ) S E E = n Σπ.μ ( ΕΕ ; ) . i = 1 ...
Page 112
... u - integrable on S if there is a sequence { f } of u - integrable simple functions converging to fin u - measure on S and satisfying in addition the equation lim f ̧ \ fm ( s ) — fn ( s ) | v ( μ , ds ) m , n = 0 . Such a sequence of u ...
... u - integrable on S if there is a sequence { f } of u - integrable simple functions converging to fin u - measure on S and satisfying in addition the equation lim f ̧ \ fm ( s ) — fn ( s ) | v ( μ , ds ) m , n = 0 . Such a sequence of u ...
Page 180
... u - integrable . Since every real measurable function is the difference of two non- negative measurable functions , it follows from the theorem that [ * ] √2g ( s ) λ ( ds ) ... integrable 180 III.10.6 III . INTEGRATION AND SET FUNCTIONS.
... u - integrable . Since every real measurable function is the difference of two non- negative measurable functions , it follows from the theorem that [ * ] √2g ( s ) λ ( ds ) ... integrable 180 III.10.6 III . INTEGRATION AND SET FUNCTIONS.
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