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 o - field Σ , and the measure space ( S , Σ * , μ ) is the Lebesgue extension of the measure space ( S , E , μ ) . Expressions such as u ...
... extension of μ . The o - field Σ * is known as the Lebesgue extension ( relative to μ ) of the o - field Σ , and the measure space ( S , Σ * , μ ) is the Lebesgue extension of the measure space ( S , E , 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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ