Linear Operators: General theory |
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Page 132
... extension theorem of Hahn and similar extension theorems , of importance in later applications , will be discussed in this section . Finally , it is shown how the extension theorems may be used to construct the classical measures of ...
... extension theorem of Hahn and similar extension theorems , of importance in later applications , will be discussed in this section . Finally , it is shown how the extension theorems may be used to construct the classical measures of ...
Page 136
... extension of μ to the o - field Zo determined by 2. Suppose that is another such extension . If μ is o - finite on to prove the uniqueness of the extension , it will suffice to show that μ ( E ) = ( E ) for every set E in 2 contained in ...
... extension of μ to the o - field Zo determined by 2. Suppose that is another such extension . If μ is o - finite on to prove the uniqueness of the extension , it will suffice to show that μ ( E ) = ( E ) for every set E in 2 contained in ...
Page 143
... extension of μ . The o - field Σ * is known as the Lebesgue extension ( relative to μ ) of the σ - field Σ , and the measure space ( S , Σ * , μ ) is the Lebesgue extension of the measure space ( S , Σ , μ ) . Expressions such as u ...
... extension of μ . The o - field Σ * is known as the Lebesgue extension ( relative to μ ) of the σ - field Σ , and the measure space ( S , Σ * , μ ) is the Lebesgue extension of the measure space ( S , Σ , μ ) . Expressions such as u ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ