Linear Operators: General theory |
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Page 143
... Borel measure in [ a , b ] . The Lebesgue measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure ...
... Borel measure in [ a , b ] . The Lebesgue measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure ...
Page 341
... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy sequence if and only if it is bounded and the limit lim , un ( e ) = u ( e ) exists for every Borel set e , in which ...
... Borel sets in S. Prove that ( i ) rca ( S ) is weakly complete . ( ii ) A sequence { u } in rca ( S ) is a weak Cauchy sequence if and only if it is bounded and the limit lim , un ( e ) = u ( e ) exists for every Borel set e , in which ...
Page 516
... Borel sets of S which is not identically zero and is p - invariant for every фє G. n - 1 42 Show that in Exercise 38 the set function u is unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a ...
... Borel sets of S which is not identically zero and is p - invariant for every фє G. n - 1 42 Show that in Exercise 38 the set function u is unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a ...
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