## Linear Operators: General theory |

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Page 108

A ^-

the form n / = I *iXE, ' i-1 where E( = i = 1, . . ., n, are disjoint sets in E with union S

and where xf = 0 if v(fi, £, ) = oo. The phrases "/^-integrable /t-

A ^-

**simple function**is fi-integrable if it differs by a null function from a function ofthe form n / = I *iXE, ' i-1 where E( = i = 1, . . ., n, are disjoint sets in E with union S

and where xf = 0 if v(fi, £, ) = oo. The phrases "/^-integrable /t-

**simple function**" ...Page 165

Since a /ij-

measurable function is //-measurable. If / is a /tj-integrable

evident that / is also a //-integrable

fi(ds) ...

Since a /ij-

**simple function**is clearly //-simple, it follows immediately that a /^-measurable function is //-measurable. If / is a /tj-integrable

**simple function**, it isevident that / is also a //-integrable

**simple function**, and that J" Ef(s)fa(ds) = J Ef{s)fi(ds) ...

Page 322

We now proceed to develop a theory of integration of scalar

respect to the vector measure /x. ... A scalar valued

this is ...

We now proceed to develop a theory of integration of scalar

**functions**withrespect to the vector measure /x. ... A scalar valued

**function**/ defined on S is fi-**simple**if it is a finite linear combination of characteristic**functions**of sets in 27*;this is ...

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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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero