## Linear Operators: General theory |

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Page 143

Then the function fx with domain E* is known as the

The cr-field E* is known as the

, and the

Then the function fx with domain E* is known as the

**Lebesgue**extension of fi.The cr-field E* is known as the

**Lebesgue**extension (relative to yu) of the a-field E, and the

**measure**space (S, E*, /n) is the**Lebesgue**extension of the**measure**...Page 188

It is clear that the measure has the property 71 = 1 ^PEt) = f[fa(Ei), E,«27(, i-i <-i

and it follows from the uniqueness part of ... 2 and its Corollary 6 is obtained by

taking (&,, Łt, pt) to be the Borel-

.

It is clear that the measure has the property 71 = 1 ^PEt) = f[fa(Ei), E,«27(, i-i <-i

and it follows from the uniqueness part of ... 2 and its Corollary 6 is obtained by

taking (&,, Łt, pt) to be the Borel-

**Lebesgue measure**on the real line for t = 1, . . ., n.

Page 223

Show that the Lebesgue-Stieltjes integral \j{m)dh{i(8)) exists and is equal to /. 6

Show that a monotone increasing function / defined on an interval is

differentiable almost everywhere with respect to

may vanish ...

Show that the Lebesgue-Stieltjes integral \j{m)dh{i(8)) exists and is equal to /. 6

Show that a monotone increasing function / defined on an interval is

differentiable almost everywhere with respect to

**Lebesgue measure**and that /'may vanish ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### 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