## Linear Operators: General theory |

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23

additive and bounded.

Show that E* contains every Borel set. 24

topological ...

23

**Suppose**that S is a metric space, that E is a ff-field, and that ft is eountablyadditive and bounded.

**Suppose**that every continuous function is //-measurable.Show that E* contains every Borel set. 24

**Suppose**that S is a compacttopological ...

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I*-»|2« 22

exists a finite constant K such that for / in CBV, \{Snj){x)\ ^K(v(f, [0, 2.t])+ sup \f(x)\),

0 g x =g 2x. 23

I*-»|2« 22

**Suppose**that (Snf)(x) -+ f(x) uniformly for every / in AC. Show that thereexists a finite constant K such that for / in CBV, \{Snj){x)\ ^K(v(f, [0, 2.t])+ sup \f(x)\),

0 g x =g 2x. 23

**Suppose**that (i) (S„f)(x) -*□ j(x) uniformly in x for / in AC. (ii) The ...Page 570

Conversely, let ft e a{f(T)), and

f)-/*)"1 belongs to &(T). By Theorem 10, h(T)(f(T)—ftI) = /. which contradicts the

assumption that fteo(f(T)). Q.E.D. ♢ 12 Theorem. Let f be in &(T), g be in &~(f(T)), ...

Conversely, let ft e a{f(T)), and

**suppose**that ft 4 f(a(T)). Then the function h(£) = (/(f)-/*)"1 belongs to &(T). By Theorem 10, h(T)(f(T)—ftI) = /. which contradicts the

assumption that fteo(f(T)). Q.E.D. ♢ 12 Theorem. Let f be in &(T), g be in &~(f(T)), ...

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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