## Linear Operators: General theory |

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Page 143

The Lebesgue measurable sets in [a, b] are the sets in the Lebesgue extension (

relative to Borel measure) of the cr-field of

Lebesgue-Stieltjes measure determined by a function / of bounded variation on a

finite ...

The Lebesgue measurable sets in [a, b] are the sets in the Lebesgue extension (

relative to Borel measure) of the cr-field of

**Borel sets**in [a, b]. Similarly, theLebesgue-Stieltjes measure determined by a function / of bounded variation on a

finite ...

Page 341

Show that a set K C ba(S, E) is conditionally compact if and only if (i) K is

bounded. (ii) There is a ... the regular countably additive set functions on the field

of

rca(S) ...

Show that a set K C ba(S, E) is conditionally compact if and only if (i) K is

bounded. (ii) There is a ... the regular countably additive set functions on the field

of

**Borel sets**in S. Prove that (i) rca(S) is weakly complete. (ii) A sequence {//„} inrca(S) ...

Page 492

In the following, St) denotes the field of

by the closed sets of S. If p is a function on 3$ with values in a /J-spacc, then as in

Definition IV. 10.3, the symbol ||jh|;(E) denotes the semi-variation of p over E e ...

In the following, St) denotes the field of

**Borel sets**in S. i.e., the <;-field generatedby the closed sets of S. If p is a function on 3$ with values in a /J-spacc, then as in

Definition IV. 10.3, the symbol ||jh|;(E) denotes the semi-variation of p over E e ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact