## Linear Operators: General theory |

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Page 164

Since the extension of a

follows that {jj,n(E)} is abounded non-decreasing set of real numbers for each E e

Ev We define X^E) = lim„ p,n{E), E e Zv By Corollary 4, Xx is countably additive ...

Since the extension of a

**non**-**negative**set function on 27 to 2^ is**non**-**negative**, itfollows that {jj,n(E)} is abounded non-decreasing set of real numbers for each E e

Ev We define X^E) = lim„ p,n{E), E e Zv By Corollary 4, Xx is countably additive ...

Page 179

Let (S, Z, p) be a positive measure space, f a

function defined on S and X(E)=jBf(s)/l{ds), EeZ. Let g be a

measurable function defined on S. Then fg is /^-measurable, and jEg(s)X(ds)=jEf(

s)g(s)/i(ds), ...

Let (S, Z, p) be a positive measure space, f a

**non**-**negative**(i-measurablefunction defined on S and X(E)=jBf(s)/l{ds), EeZ. Let g be a

**non**-**negative**X-measurable function defined on S. Then fg is /^-measurable, and jEg(s)X(ds)=jEf(

s)g(s)/i(ds), ...

Page 314

By part (b), v(U(fa), E) = \V{M\ = |^| = vfa, Q.E.D. 12 Theorem. .4 subset K of ba(S,

Z) is weakly sequentially compact if and only if there exists a

(S, Z) such that lim X(E) = 0 uniformly for XeK. Proof. Let K Q ba(S, Z) be weakly ...

By part (b), v(U(fa), E) = \V{M\ = |^| = vfa, Q.E.D. 12 Theorem. .4 subset K of ba(S,

Z) is weakly sequentially compact if and only if there exists a

**non**-**negative**u in ba(S, Z) such that lim X(E) = 0 uniformly for XeK. Proof. Let K Q ba(S, Z) be weakly ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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