By A.B. Kharazishvili

This ebook is dedicated to a couple effects from the classical aspect Set conception and their purposes to sure difficulties in mathematical research of the true line. realize that a variety of subject matters from this conception are offered in numerous books and surveys. From one of the most vital works dedicated to element Set conception, allow us to to begin with point out the wonderful booklet through Oxtoby [83] during which a deep analogy among degree and classification is mentioned intimately. extra, a fascinating normal method of difficulties pertaining to degree and type is built within the recognized monograph through Morgan [79] the place a basic inspiration 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 e-book, yes sessions of subsets of the genuine line are studied and diverse cardinal valued features (characteristics) heavily attached with these periods are investigated. evidently, the IT-ideal of all Lebesgue degree 0 subsets of the true line and the IT-ideal of all first class subsets of an identical line are commonly studied in [19], and several other rather new effects relating this subject are awarded. eventually, it's moderate to note right here that a few precise units of issues, the so-called singular areas, are thought of within the classi

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**Sample text**

Only two cases are possible. 1. There exists an element y of Y such that card({{ : y E Xe}) = WI. In this case, we may suppose, without loss of generality, that ('*
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We also want to remark that eonditions 1), 2) and :3) imply the following fact: the completion of the given measure v is an extension of some measure proportional to the dassieal Lebesgue measure on the real line R. NONMEASURABLE SETS 41 A similar argument shows us that no Vitali subset of R has the Baire property with respect to the standard topology of R. Namely, we can formulate the following result. Theorem 2. Let f be a countable subgroup of the additive group of the real line and let T be a topology on R, such that 1) the pair (R, T) is a topological group; 2) R is not a first category space with respect to T; 3) f is a non discrete subgroup of (R, T).

Let us define an element z of E by the following formula: zE and z{ = 0 if a < ~ < WI. Obviously, we have = YE if ~ ~ a, z E f, Therefore f z E n{Wn : n < w}. n (n{Wn : n < w}) =f 0 and, consequently, f n (n{Vn : n < w}) =f 0. So we obtained a contradiction which shows that the set r does not have the Baire property in E. Let us emphasize once more that this fact is proved in the theory ZF & DC. Now, let us make some simple remarks concerning a Haar measure given on E. Actually, since E is a compact topological group, there exists a probability Haar measure It on E.