By John C. Oxtoby

In this version, a collection of Supplementary Notes and comments has been additional on the finish, grouped in line with bankruptcy. a few of these name realization to next advancements, others upload additional clarification or extra comments. lots of the comments are followed by means of a in brief indicated evidence, that's occasionally various from the single given within the reference mentioned. The checklist of references has been elevated to incorporate many contemporary contributions, however it continues to be no longer meant to be exhaustive. John C. Oxtoby Bryn Mawr, April 1980 Preface to the 1st version This ebook has major topics: the Baire type theorem as a style for proving life, and the "duality" among degree and classification. the class procedure is illustrated through a number of ordinary functions, and the analogy among degree and classification is explored in all of its ramifications. To this finish, the weather of metric topology are reviewed and the primary homes of Lebesgue degree are derived. It seems that Lebesgue integration isn't really crucial for current purposes-the Riemann necessary is adequate. recommendations of normal degree conception and topology are brought, yet not only for the sake of generality. take into account that, the time period "category" refers regularly to Baire class; it has not anything to do with the time period because it is utilized in homological algebra.

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**Example text**

8. The set of points of discontinuity of any monotone function f is countable. Any countable set is the set of points of discontinuity of some monotone function. Proof. If f is monotone, there can be at most If(b) - f(a)IIB points in (a, b) where w(x) ~ B. Hence the set of points of discontinuity of f is countable. On the other hand, let {Xi} be any countable set, and let LBi be a convergent series of positive real numbers. The function f(x) = LXi:5xBi is a monotone bounded function. It has the property that w(x i) ,; Bi for each i, and w(x) = 0 for all X not in the sequence Xi· 0 This should be compared with the much deeper theorem, due to Lebesgue, that any monotone function is differentiable (has a finite derivative) except at a set of points of measure zero [31, p.

For any n ~ ni , we have m(E t;,. En) = m((E t;,. Fi ) t;,. (En; t;,. En)) ~ m(E t;,. Fi ) + m(En; t;,. En)":::; 1/2i- 1 + 1/2i . En converges to E in (S, (l). We remark that when m is taken to be 2-dimensional Lebesgue measure in the plane, the space (S, (l) contains a subset isometric to (It (1). For any real-valued function f on [a, b], let 4J(f) denote its ordinate set, that is, the set It follows that 4J(f)={(x,y):a~x~b, O~y~f(x) or f(x)~y~O}. It is not hard to see that if f and g belong to R [a, b], then J~lf - gl = m(4J(j)t;,.

Proof. Let f1,f2, ... be a winning strategy for (B). Let 10 denote the interior of any interval I. Given f1' it is possible to define a sequence of closed intervals Ji (i = 1,2, ... ) contained in Ig such that (i) the intervals Ki = f1 (10' JJ are disjoint, and (ii) the union oftheir interiors is dense in 10 , One way to do this is as follows. Let S be a sequence consisting of all closed intervals that have rational endpoints and are contained in Ig. Let J 1 be the first term of S. Having defined J 1 , ...