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Question: in this example you will construct a subset of 01...

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In this example you will construct a subset of 0,1 that is not measurable with respect to the Lebesgue measure. (Therefore not all subsets of [0, 1 are Borel.) Let C denote the one-dimensional Lebesgue measure such that C(a, b)-b- a. Let be the equivalence relation defined as follows: x~y provided s-yEQ Note that any two equivalence classes are either disjoint or the coincide, and [0,1 can be written as the disjoint union of all equivalence classes: for some index set I (not necessarily countable). Now, choose exactly one element ra from each E., and let N-{ra : αΕΊ) (a) Let r be an enumeration of all the rationals in -1, 1], and consider the translates N NTk. Show that the sets N are disjoint, and に! (b) Assume N is measurable, and prove that (c) Conclude that the estimates above yield a contradiction; hence, N is not mea- surable.

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