## Linear Operators: General theory |

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Page 108

A /i-

the form Tl / = 2 1 '.*£. □ f— 1 where 2J, = f~l(xt), i = 1, . . ., n, are disjoint sets in E

with union .V and where xt = 0 if v(fi, Et) = oo. The phrases "//-integrable ^-simpk ...

A /i-

**simple function**is [t-integrable if it differs by a null function from a function ofthe form Tl / = 2 1 '.*£. □ f— 1 where 2J, = f~l(xt), i = 1, . . ., n, are disjoint sets in E

with union .V and where xt = 0 if v(fi, Et) = oo. The phrases "//-integrable ^-simpk ...

Page 165

Since a /tj-

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

evident that / is also a «-integrable

fi(ds) ...

Since a /tj-

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

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

**simple function**, and that J"£ f(s)fi1{ds) = f£ f(s)fi(ds) ...

Page 322

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

respect to the vector measure //. A fi-null set is ... 0.1 4 that / is //-measurable if

and only if it is the limit /t-almost everywhere of a sequence of //-

By III. 6.

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

respect to the vector measure //. A fi-null set is ... 0.1 4 that / is //-measurable if

and only if it is the limit /t-almost everywhere of a sequence of //-

**simple functions**.By III. 6.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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