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Page 106
The functions totally u-measurable on S, or, if // is understood, totally measurable
on S are the functions in the closure TM(S) in F(S) of the //-simple functions. If for
every E in E with v(fi, E) < oo, the product of / with the characteristic function %E ...
The functions totally u-measurable on S, or, if // is understood, totally measurable
on S are the functions in the closure TM(S) in F(S) of the //-simple functions. If for
every E in E with v(fi, E) < oo, the product of / with the characteristic function %E ...
Page 179
measurable function defined on S and l(E)=jEf(s)v(ds), EeZ. Let g be a non-
negative X-measurable function defined on S. Then fg is u- measurable, and j£ g(
s)k(ds) = jE f(s)g(s)fi(ds), E e E. Proof. The ,u-measurability of fg follows from
Lemma ...
measurable function defined on S and l(E)=jEf(s)v(ds), EeZ. Let g be a non-
negative X-measurable function defined on S. Then fg is u- measurable, and j£ g(
s)k(ds) = jE f(s)g(s)fi(ds), E e E. Proof. The ,u-measurability of fg follows from
Lemma ...
Page 180
Since every real measurable function is the difference of two non- negative
measurable functions, it follows from the theorem that [*] j£ g(s)l(ds) = jE f(s)g(s)t*(
ds), E e Z, for a real measurable function g. Since the real and imaginary parts of
a ...
Since every real measurable function is the difference of two non- negative
measurable functions, it follows from the theorem that [*] j£ g(s)l(ds) = jE f(s)g(s)t*(
ds), E e Z, for a real measurable function g. Since the real and imaginary parts of
a ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero