Linear Operators: General theory |
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Page 186
... measure μ of Lemma 1 is countably additive on Ø . Consequently , the restriction of u to the o - field generated by ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and s2 in S2 the set E ( $ 2 ) = { $ 1 ...
... measure μ of Lemma 1 is countably additive on Ø . Consequently , the restriction of u to the o - field generated by ... positive measure spaces ( S1 , 21 , μ1 ) and ( S2 , 22 , 2 ) . For each E in E and s2 in S2 the set E ( $ 2 ) = { $ 1 ...
Page 212
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
... positive measure defined on the o - field of Borel sets of a compact metric space S. A set ACS is said to be covered in the sense of Vitali by a family F of closed sets if each Fe F has positive μ - measure and there is a positive ...
Page 725
... measure . ( Hint . Consider the map f ( s ) → ZA ( S ) f ( 98 ) for each Ae with u ( A ) < ∞ . ) € 37 Let ( S , 2 , μ ) be a positive measure space , and T a non - nega- tive linear transformation of L1 ( S , E , μ ) into itself ...
... measure . ( Hint . Consider the map f ( s ) → ZA ( S ) f ( 98 ) for each Ae with u ( A ) < ∞ . ) € 37 Let ( S , 2 , μ ) be a positive measure space , and T a non - nega- tive linear transformation of L1 ( S , E , μ ) into itself ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ