By A. N. Kolmogorov

Reprinted the following in a single quantity, the 1st half is dedicated to metric and general areas. starting with a quick creation to set concept and mappings, the authors supply a transparent presentation of the speculation of metric and whole metric areas. the main of contraction mappings and its purposes to the evidence of lifestyles theorems within the concept of differential and fundamental equations gets precise research, as do non-stop curves in metric areas — a subject seldom mentioned in textbooks.

Part One additionally contains discussions of different topics, comparable to components of the idea of normed linear areas, susceptible sequential convergence of components and linear functionals, adjoint operators, and linear operator equations. half specializes in an exposition of degree thought, the Lebesque period and Hilbert house. either elements characteristic a variety of routines on the finish of every part and contain worthy lists of symbols, definitions, and theorems.

**Read or Download Elements of the Theory of Functions and Functional Analysis, Volume 2, Measure. The Lebesgue Integral. Hilbert Space PDF**

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**Additional info for Elements of the Theory of Functions and Functional Analysis, Volume 2, Measure. The Lebesgue Integral. Hilbert Space**

**Sample text**

Since 0 = 0 u 0, it follows that The following two theorems on measures in semi-rings will be applied repeatedly in the sequel. THEOREM 1. Let be a measure defined on a semi-ring If REMARK. A1, • A •, E where the sets Ak are pairwise disjoint subsets of A, then /1(A). Proof. Since is a semi-ring, in view of Lemma 1 of §34 there exists a partition A= (s n, Ak E in which the first n sets coincide with the given sets A1, the measure of an arbitrary set is nonnegative, An.. Since Ek=l/1(Ak) Ek=l/1(Ak) = /1(A).

3. a) It follows from Ex. 2 and the text that any triangle is urable and that its is the classical area. b) Show, therefore, that a regular polygon is and receives its classical area. 28 [cH. , a closed disk, is 4. Show that the plane set A = not j(2) -measurable. { (x, y) x2 + y2 1, x, y rationall is §37. Complete additivity. The general problem of the extension of measures It is often necessary to consider countable unions as well as finite unions. Therefore, the condition of additivity we imposed on a measure (§34, Def.

This neighborhood has positive measure; hence /2{x:J'(x) g(x)} > 0, that is, the continuous functions f and g cannot be equivalent if they differ even at a single point. Obviously, the equivalence of two arbitrary measurable (that is, in general, discontinuous) functions does not imply their equality; for instance, the function equal to 1 at the rational points and 0 at the irrational points is equivalent to the function identically zero on the real line. THEOREM 9. A function f(x) defined on a measurable set E and equivalent on E to a measurable function g(x) is measurable.