## Linear Operators: General theory |

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Page 108

A ^-

the form n / = 2 XiXEi ' where Et = f~l{xt), i = 1, . . ., n, are disjoint sets in X with

union S and where x( = 0 if v(fi, E{) — oo. The phrases "^-integrable //-simple ...

A ^-

**simple function**is fi-integrable if it differs by a null function from a function ofthe form n / = 2 XiXEi ' where Et = f~l{xt), i = 1, . . ., n, are disjoint sets in X with

union S and where x( = 0 if v(fi, E{) — oo. The phrases "^-integrable //-simple ...

Page 165

Since a /^-

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

evident that / is also a yu-integrable

fi(ds) ...

Since a /^-

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

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

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

Page 322

We now proceed to develop a theory of integration of scalar

respect to the vector measure ft. ... A scalar valued

is ...

We now proceed to develop a theory of integration of scalar

**functions**withrespect to the vector measure ft. ... A scalar valued

**function**/ defined on S is fx-**simple**if it is a finite linear combination of characteristic**functions**of sets in Z*; thisis ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations disjoint Doklady Akad Duke Math element ergodic theorem exists finite dimensional function defined Hausdorff space Hence Hilbert space homeomorphism ibid inequality integral Lebesgue Lebesgue measure Lemma linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc Proof properties proved real numbers reflexive Riesz Russian Sbornik N. S. scalar self-adjoint semi-group sequentially compact Show simple functions spectral Studia Math subset subspace Suppose theory topological space Trans uniformly Univ valued function Vber vector space weak topology weakly compact zero