By A.B. Kharazishvili

This ebook is dedicated to a couple effects from the classical aspect Set thought and their functions to sure difficulties in mathematical research of the true line. detect that numerous issues from this concept are provided in numerous books and surveys. From one of the most vital works dedicated to element Set concept, allow us to firstly point out the wonderful publication by way of Oxtoby [83] within which a deep analogy among degree and type is mentioned intimately. additional, a fascinating common method of difficulties pertaining to degree and type is built within the famous monograph by means of Morgan [79] the place a basic proposal of a class base is brought and investigated. We additionally desire to point out that the monograph through Cichon, W«;glorz and the writer [19] has lately been released. In that publication, yes periods of subsets of the genuine line are studied and numerous cardinal­ valued services (characteristics) heavily attached with these periods are investigated. evidently, the IT-ideal of all Lebesgue degree 0 subsets of the genuine line and the IT-ideal of all first classification subsets of an analogous line are broadly studied in [19], and several other fairly new effects pertaining to this subject are provided. eventually, it truly is moderate to note right here that a few targeted units of issues, the so-called singular areas, are thought of within the classi

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Extra resources for Applications of Point Set Theory in Real Analysis

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The existence of a bijection g : A -+ B with the graph contained in the (2 - 2)-correspondence G (defined above) implies, in the theory ZF & DC, the existence of a Lebesgue nonmeasurable function acting from R into R. ount of the result of Solovay mentioned in Remark 1, we conclude that (1) the Hall theorem cannot be proved in the theory ZF & DC; (2) Theorem 5 cannot be proved in the theory ZF & DC. We also have the next fact: 46 2 (3) it cannot be proved, in the theory ZF & DC, that there exists a linear ordering of the Vitali partition {V; : i E I}.

Clearly, Q is a subgroup of the additive group of R. Let us consider a binary relation G c R x R defined by the following formula: (x,Y)EG~X-YEQ. It is easy to see that G is an equivalence relation on the real line. The graph of this relation is a very simple subset of the Euclidean plane R 2 . Namely, it can be represented as the union of a countable family of straight lines lying in R 2 and parallel to the line {(x,y) E R2 : x = y}. Let us denote by {V; : i E l} the partition of R canonically associated with G.

Notiee that the argument presented above also proves a more general statement. In order to formulate it, we need the notion of a set of Vitali type. Let r be a subgroup of the additive group of the real line R. ally assoeiated with the equivalenee relation x ER & y ER & x - y E r. Let X be any selector of this partition. We shall say that X is a r -selector (or that X is a set of Vitali type with respect to the group r). It ean easily be seen that the preeeding argument establishes the following result.

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