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 { u ( 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 { u ( IB ) } is an increasing sequence ) . To see that μ is countably additive ...
Page 341
... set S , and let 1 be the o - field generated by Σ . Let μ be a non - negative finite countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy ...
... set S , and let 1 be the o - field generated by Σ . Let μ be a non - negative finite countably additive measure ... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy ...
Page 634
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have 1 + 00 • + ∞ + ∞ [ +2 [ + % ( E ) ( s , t ) ds dt = [ + ] % ( st ) dsdt 00 -∞ • + 00 00 = [ ( [ ZE ( 8-1 ) ds ) dt ...
... Borel set if E is a Borel set . Now let E be a Borel set of measure zero . By Fubini's theorem ( III.11.9 ) , we have 1 + 00 • + ∞ + ∞ [ +2 [ + % ( E ) ( s , t ) ds dt = [ + ] % ( st ) dsdt 00 -∞ • + 00 00 = [ ( [ ZE ( 8-1 ) ds ) dt ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ