## Linear Operators: General theory |

### From inside the book

Results 1-3 of 80

Page 151

Hence, g = / almost

Corollary. (Lebesgue dominated convergence theorem) Let I ^ p < co, let (S, E, ix)

bea measure space, and let {/„} be a sequence of functions in LP(S, E, fx, ...

Hence, g = / almost

**everywhere**, and consequently / e Lv and \U-t\, -0. Q.E.D. 16Corollary. (Lebesgue dominated convergence theorem) Let I ^ p < co, let (S, E, ix)

bea measure space, and let {/„} be a sequence of functions in LP(S, E, fx, ...

Page 722

limA+00 (XR(X; A)j)(s) exists almost

is a topological space, and if, for each A > 0, R(X; A )f is continuous if / is

continuous, then for 1 ^ p < oo and / e Lp the limit lim^o (XR(X; A)f)(s) exists

almost ...

limA+00 (XR(X; A)j)(s) exists almost

**everywhere**for feLv, 1 gp < a. Show that if Sis a topological space, and if, for each A > 0, R(X; A )f is continuous if / is

continuous, then for 1 ^ p < oo and / e Lp the limit lim^o (XR(X; A)f)(s) exists

almost ...

Page 726

constant ^-measurable function / can satisfy f(q>s) = /(*) fi -almost

and that in this case, any //-measurable function g satisfying g(<ps) = Xg(s) //-

almost

that if ...

constant ^-measurable function / can satisfy f(q>s) = /(*) fi -almost

**everywhere**,and that in this case, any //-measurable function g satisfying g(<ps) = Xg(s) //-

almost

**everywhere**, where \X\ = 1, satisfies \g(s)\ = 1 //-almost**everywhere**. Showthat if ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

### 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