## Linear Operators: General theory |

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Page 106

The

on S are the

every E in E with v(fi, E) < oo, the product of / with the characteristic

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 forevery E in E with v(fi, E) < oo, the product of / with the characteristic

**function**%E ...Page 179

negative X-

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

ds), E e Z, for a real

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 ofa ...

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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