## Linear Operators: General theory |

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Page 140

It can also be extended by defining the measure differently on the basic field E.

We shall illustrate this latter remark by describing here the construction of a

Radon measure on an

It can also be extended by defining the measure differently on the basic field E.

We shall illustrate this latter remark by describing here the construction of a

Radon measure on an

**interval**. 15 Definition. An**interval**is a set of points in the ...Page 141

The argument may be repeated by defining e3 — an>— c and choosing

appropriate points in the

integer k = 1, 2, . . ., there are points at, bt with c < a„ ^ 6n ^ . . . iŁ ax ^ 6X ^ c+ex

such that 2 ...

The argument may be repeated by defining e3 — an>— c and choosing

appropriate points in the

**interval**(c, 0+63]- By induction, it is clear that for everyinteger k = 1, 2, . . ., there are points at, bt with c < a„ ^ 6n ^ . . . iŁ ax ^ 6X ^ c+ex

such that 2 ...

Page 223

5 Let Abea function of bounded variation on the

the right. Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes

integral / = jlg(s)dh(s) exists. Let / be a continuous increasing function on an open

...

5 Let Abea function of bounded variation on the

**interval**(a, b) and continuous onthe right. Let g be a function defined on (a, b) such that the Lebesgue-Stieltjes

integral / = jlg(s)dh(s) exists. Let / be a continuous increasing function on an open

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

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Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero