## Linear Operators: General theory |

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Page 108

A ^-

the form n / = 2 Xi%E< > 1-1 where Et = f~l(xt), i = 1, . . ., n, are disjoint sets in £

with union S and where ,r, = 0 if v(/u, E() = oo. The phrases "/i-intcgrable ^u-

simple ...

A ^-

**simple function**is fi-integrable if it differs by a null function from a function ofthe form n / = 2 Xi%E< > 1-1 where Et = f~l(xt), i = 1, . . ., n, are disjoint sets in £

with union S and where ,r, = 0 if v(/u, E() = oo. The phrases "/i-intcgrable ^u-

simple ...

Page 165

Since a /^-

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

evident that / is also a /i-integrable

ds) for ...

Since a /^-

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

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

**simple function**, and that $Ef(s)fa(ds)=$Ef(s)fi(ds) for ...

Page 322

We now proceed to develop a theory of integration of scalar functions with

respect to the vector measure //. ... If / is the //-

, ...,£„ are sets in 27, then the integral of / over a set E e 27 is defined by the

equation It ...

We now proceed to develop a theory of integration of scalar functions with

respect to the vector measure //. ... If / is the //-

**simple function**2<=i ai£i(> where E(, ...,£„ are sets in 27, then the integral of / over a set E e 27 is defined by the

equation It ...

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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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### Common terms and phrases

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