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 u 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 u is countably additive ...
Page 341
... set S , and let 1 be the o - field generated by 2. Let μ be a non - negative finite μ 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 2. Let μ be a non - negative finite μ 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 • + ∞ + ∞0 [ ** S ** % , ( E ) ( s , t ) ds dt = [ ** S *** ZE ( 8—1 ) dsdt Sto -∞ -∞ + ∞ = -∞ | | XE ( 81 ) ds ...
... 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 • + ∞ + ∞0 [ ** S ** % , ( E ) ( s , t ) ds dt = [ ** S *** ZE ( 8—1 ) dsdt Sto -∞ -∞ + ∞ = -∞ | | XE ( 81 ) ds ...
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