## Linear Operators, Part 1 |

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Page 106

The

on S are the

every E in X with v(u, E) < 00, the product Zef of f with the characteristic

The

**functions**totally u-**measurable**on S, or, if u is understood, totally**measurable**on S are the

**functions**in the closure TM (S) in F(S) of the u-simple**functions**. If forevery E in X with v(u, E) < 00, the product Zef of f with the characteristic

**function**...Page 179

Let (S, 2, u) be a positive measure space, f a nonnegative u-

defined on S and Ž(E) = J. f(s)u (ds), Ee 2. Let g be a non-negative A-

Let (S, 2, u) be a positive measure space, f a nonnegative u-

**measurable function**defined on S and Ž(E) = J. f(s)u (ds), Ee 2. Let g be a non-negative A-

**measurable****function**defined on S. Then fg is su-measurable, and s, g(s)2(ds)=|f(s)p(s)a(ds) ...Page 180

Since every real

ds), Be2, for a real

...

Since every real

**measurable function**is the difference of two nonnegative**measurable functions**, it follows from the theorem that *] s, g(s)2(ds)=s, f(s)p(s)a(ds), Be2, for a real

**measurable function**g. Since the real and imaginary parts of a...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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### Common terms and phrases

additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero