## Linear Operators, Part 1 |

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Page 106

The functions totally

on S are the functions in the closure TM (S) in F(S) of the u-simple functions. If for

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

The functions totally

**u**-**measurable**on S, or, if u is understood, totally measurableon S are the functions in the closure TM (S) in F(S) of the u-simple functions. If for

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

Page 119

(b) the function g defined by g(s) = f(s) if s 6 St. US-, g(s) = 0 if s e S+ US-, is

valued) which is defined only on the complement of a u-null set NC S. Then we

say ...

(b) the function g defined by g(s) = f(s) if s 6 St. US-, g(s) = 0 if s e S+ US-, is

**u**-**measurable**. Next suppose that we consider a function f (vector or extended real-valued) which is defined only on the complement of a u-null set NC S. Then we

say ...

Page 178

(s)f(s)g(s), se S, and the pointwise limit of a sequence of

and shall assume that v(

is ...

(s)f(s)g(s), se S, and the pointwise limit of a sequence of

**measurable**functions is**measurable**, it will suffice then to show that ze.jg is**measurable**. Thus we mayand shall assume that v(

**u**, F) < 0 and v(2, F) < 00. Since g is A-**measurable**thereis ...

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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

### 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