Linear Operators: General theory |
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Page 144
... Borel set B. Let In = { s | −n < s < + n } , and put μ ( B ) lim ( IB ) for every Borel set ( the limit exists as an extended positive real number , since { μ ( IB ) } is an increasing sequence ) . To see that μ is countably additive ...
... Borel set B. Let In = { s | −n < s < + n } , and put μ ( B ) lim ( IB ) for every Borel set ( the limit exists as an extended positive real number , since { μ ( IB ) } is an increasing sequence ) . To see that μ is countably additive ...
Page 215
... set 4 , , Q , we see there is a Borel set BCQ such that A ,, Q - B is a μ - null set and i + 1 λ ( B ) ≥ μ ( B ) . ว Since A , CBU ( A ,, — Q ) U ( A ,, Q — B ) , μ ( B ) ≥ ß . Thus i ( i + 1 ) ( B + ɛ ) > 2 ( Q ) ≥ 2 ( B ) > β j ...
... set 4 , , Q , we see there is a Borel set BCQ such that A ,, Q - B is a μ - null set and i + 1 λ ( B ) ≥ μ ( B ) . ว Since A , CBU ( A ,, — Q ) U ( A ,, Q — B ) , μ ( B ) ≥ ß . Thus i ( i + 1 ) ( B + ɛ ) > 2 ( Q ) ≥ 2 ( B ) > β j ...
Page 341
... set S , and let be the o - field generated by E. Let u be a non - negative finite 1 countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { μ } in rca ( S ) is a weak Cauchy sequence ...
... set S , and let be the o - field generated by E. Let u be a non - negative finite 1 countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { μ } in rca ( S ) is a weak Cauchy sequence ...
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