PROBLEMS AND EXERCISES IN INTEGRAL EQUATIONS

somit guiwollel sit PRELIMINARY REMARKS (x) 1. Measurable sets. Let E be some set of points of an nterval S = [a, b]. Denote the complement of E with respect o S by Cr; i.e., by definition CE consists of points which do not belong to E. There are a variety of ways in which the points of set E may be included in a finite or countable system of intervals A1, A2, An We denote by a the sum of the lengths of the intervals For any system of intervals covering E, Χ1, A2, An Σαν 0 The lower bound of ∑α, which depends solely on the set E, is called the exterior measure and is denoted m* E. From the definition of an exterior measure it follows that for any & > 0 there exists a system of intervals a₁, a₂, which include all points of the set E such that *E<a <m*E E+ +ε The interior measure m. E of the set E is the difference between the length of the interval S and the exterior measure of the complement of the set; i.e., mEb-a-m* CE If the exterior and interior measures of E are equal, then the set E is called measurable in the sense of Lebesgue (Lebesgue measurable, or, simply, measurable), while the common value of the measures m* E and m E is called the Lebesgue measure of E (or, simply, the measure of E) and is denoted by me or mes E. The measure of the interval (a, b) is its length: mes (a, b)=b-a. The set w of points of the interval (a, b) is called a set of measure zero if w can be covered by intervals the sum of whose lengths is arbitrarily small.