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11. 35) holds for infinitely many pairs p,q but only for finitely many relatively prime ones. Then x is rational. 6). But however rapidly cp goes to infinity, A is nonempty, even uncountable. Hint. Consider x = E7_ 1 1 / 2" (k) for integral a(k) increasing very rapidly to infinity. SECTION 2. PROBABILITY MEASURES Spaces Let II be an arbitrary space or set of points w. In probability theory SZ consists of all the possible results or outcomes co of an experiment or obse rv ation. For observing the number of heads in n tosses of a coin the space f1 is (0, 1 ...

47.... 1 for n = 1, 2, .... It will be shown that U 7=o-4 • is - strictly smaller than 0 _ o(fo ). 15. * This topic may be omitted. SECTION 2. PROBABILITY MEASURES 31 If a,r and b,, are rationals decreasing to a and b, then (a, b] = U m (1 n (a m , b,,1= U m (U „(a m , bn ]`)` e J. 28) for some sequence A 1 , A 2 ,... of sets in ,4_ 1 . ),4)n_I(Am21, mz , n It follows by induction that every element of some sequence of sets in A. "), ), 1,2, .. 30) ^( A 1 A... ) = ^1( A mii, A... } U ^2( A mzi ' `4 mzz ^ ...

Regard 0 as an element of Y of length O. If A = U7 I I the It being t For the notation, see [A4] and [Al0]. 12) n A (A)= A(Ii ) = llil. As pointed out in Section 1, there is a question of uniqueness here, because A will have other representations as a finite disjoint union U. 1 J1 of 5-sets. ) The definition is indeed consistent. 12) defines a set function h on 211 , a set function called Lebesgue measure. 2. Lebesgue measure measure on the field a,. is a (countably additive) probability Suppose that A = U1 Ak, where A and the A k are 2 0 -sets and the A k are disjoint.

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