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Page 183
... follows immediately . Clearly , g ( $ ( · ) ) is a μ1 - integrable simple function if g is a μ2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ 1 8 ( 82 ) μ2 ( d $ 2 ) = √ 4-1 ...
... follows immediately . Clearly , g ( $ ( · ) ) is a μ1 - integrable simple function if g is a μ2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ 1 8 ( 82 ) μ2 ( d $ 2 ) = √ 4-1 ...
Page 403
... follows easily from ( 1 ) that ( 2 ) μ11 ( еÑ1 ) = μs ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of § a cylinder set if there exists an orthogonal projection E of ...
... follows easily from ( 1 ) that ( 2 ) μ11 ( еÑ1 ) = μs ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of § a cylinder set if there exists an orthogonal projection E of ...
Page 714
... follows from its definition that Ep is a projection , that EPED EDEP 0 , and that I Ep + ED . Further T commutes ... follows that Txp 0 and Taxp → 0 ; but since { T } has infinitely many terms equal to xp , it follows that ap = 0 ...
... follows from its definition that Ep is a projection , that EPED EDEP 0 , and that I Ep + ED . Further T commutes ... follows that Txp 0 and Taxp → 0 ; but since { T } has infinitely many terms equal to xp , it follows that ap = 0 ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ