## 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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