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Page 336
... equal u - almost everywhere if and only if ( A ( ƒ1 ) ) and 0 ( A ( f2 ) ) are equal . Now the o - finite space ( T , 21 , 0 ) can contain at most a countable number of disjoint measur- able sets of non - zero 0 - measure , and so any ...
... equal u - almost everywhere if and only if ( A ( ƒ1 ) ) and 0 ( A ( f2 ) ) are equal . Now the o - finite space ( T , 21 , 0 ) can contain at most a countable number of disjoint measur- able sets of non - zero 0 - measure , and so any ...
Page 558
... equal to one in a neighborhood of 2 , and identically equal to zero in a neighborhood of each point of o ( T ) { } ' . Put E ( 20 ) e ( T ) . The next theorem follows immediately from Theorem 5 . = 6 THEOREM . ( a ) E ( 2 ) 0 if and ...
... equal to one in a neighborhood of 2 , and identically equal to zero in a neighborhood of each point of o ( T ) { } ' . Put E ( 20 ) e ( T ) . The next theorem follows immediately from Theorem 5 . = 6 THEOREM . ( a ) E ( 2 ) 0 if and ...
Page 575
... equal to zero for u in a neighborhood of o , and equal to ( 2 - μ ) -1 for μ in a neighbor- hood of the remaining points of o ( T ) . Then g ( T ) ( 21 − T ) = ( ¿ I —T ) g ( T ) = 1 - E ( o ) . If we define A1 : X → X by A1 = AE ( σ ) ...
... equal to zero for u in a neighborhood of o , and equal to ( 2 - μ ) -1 for μ in a neighbor- hood of the remaining points of o ( T ) . Then g ( T ) ( 21 − T ) = ( ¿ I —T ) g ( T ) = 1 - E ( o ) . If we define A1 : X → X by A1 = AE ( σ ) ...
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