## Linear Operators, Part 1 |

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Page 96

A

finite family {A1, ..., A,} of disjoint subsets of t whose union is in t. For an example

of a ...

A

**set function**u defined on a family r of**sets**is said to be**additive**or finitely**additive**if u(q) = 0 and u(A1 U 42. . . U An) – pu(A1)+u(A2)+. - ..+u(An), for everyfinite family {A1, ..., A,} of disjoint subsets of t whose union is in t. For an example

of a ...

Page 97

The total variation v(u) of an

dominates u in the sense that v(u, E) > u(E) for ... 4 below by proving that v(u) is

the smallest of the non-negative

Ee 2.

The total variation v(u) of an

**additive set function**u is important because itdominates u in the sense that v(u, E) > u(E) for ... 4 below by proving that v(u) is

the smallest of the non-negative

**additive set functions**A such that A(E) = u(E) forEe 2.

Page 126

Countably

results of the preceding sections can be considerably extended. 1 DEFINITION.

Countably

**Additive Set Functions**The basis for the present section is a countably**additive set function**defined on a g-field of subsets of a set. In this case theresults of the preceding sections can be considerably extended. 1 DEFINITION.

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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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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