A bound for the representability of large numbers by ternary by Golubeva E.P.

By Golubeva E.P.

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2. The 'union' of a finite or enumerable collection of sets {xm(x)} (m = I, 2, ... ) is the set a(x) = I-{1-XI(x)}{1-x 2(x)} ... {I-xn(x)} if the collection is finite, or the set if the collection is enumerable. 4. The union of the sets x1, x2 , ••• consists of the points which belong to at least one of these sets. For a(x) = I only if one of the terms {1-xp(x)} is zero. Although we need no special symbol, such as Xm(x), for the intersection of the sets Xv x2 , ••• , it will be convenient to denote n 00 their union by U Xn(x), or by U Xn(x), or by ex u /3, if there are only two sets ex and f3.

2 (the Reine-Borel theorem). If o:(x) is a compact set of points E covered by an enumerable collection 0 of Indicators 52 open sets {o:n(x)} (n = l, 2, ... ) then E is also covered by a finite number of sets in the collection 0. Let Then the possible values of the function cfon(x) are 0, l, 2, ... , n. Hence cfon(x) has a greatest lower bound An which is certainly attained at some point gn, and cfon(x) ~ cf;,,(gn) = An for all X in E. Since E is compact, the sequence of points {gn} has at least one point of accumulation gin E, and moreover, for some integer q,

These are the familiar algebraical techniques of addition, multiplication, and their inverses, together with the analytical techniques of limiting processes. 2. The method of bracketing The two techniques of generalization characteristic of Lebesgue theory are the method of bracketing and the method of monotonic convergence. 4 as a technique for extending the concept of integration, but it is interesting to indicate the very extensive class of functionals to which it can be applied and to expose the inherent restrictions in this method.

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