## Linear Operators, Part 1 |

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Page 164

Since the extension of a

follows that {s1,(E)} is a bounded ... If A, is not purely finitely additive, there is a

non-zero 2' e ca(S, 2) such that A's 2–21; hence A, is Al-H2' = A and sup.c u(S) =

2 ...

Since the extension of a

**non**-**negative**set function on X to X, is**non**-**negative**, itfollows that {s1,(E)} is a bounded ... If A, is not purely finitely additive, there is a

non-zero 2' e ca(S, 2) such that A's 2–21; hence A, is Al-H2' = A and sup.c u(S) =

2 ...

Page 179

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

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

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

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

**nonnegative**u-measurable functiondefined on S and Ž(E) = J. f(s)u (ds), Ee 2. Let g be a

**non**-**negative**A-measurablefunction defined on S. Then fg is su-measurable, and s, g(s)2(ds)=|f(s)p(s)a(ds) ...

Page 314

equation 21(F) = ui(EF), Fe XI. Then U(21)(G) = (Uul)(GE) for G in 2. By part (b), v(

U(un), E) = U(1) = |A| = v(uh, E). Q.E.D. 12 THEOREM. A subset K of bass, 2) is

weakly sequentially compact if and only if there exists a

equation 21(F) = ui(EF), Fe XI. Then U(21)(G) = (Uul)(GE) for G in 2. By part (b), v(

U(un), E) = U(1) = |A| = v(uh, E). Q.E.D. 12 THEOREM. A subset K of bass, 2) is

weakly sequentially compact if and only if there exists a

**non**-**negative**u in bass, ...### What people are saying - Write a review

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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