By Pier, Jean-Paul

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For, let {λη} be a monotone increasing sequence, for which Σλΰ2 < °° ; by 1 . 7 . 1 it follows from this that the series Σλ7ι1Γη(χ) is convergent almost everywhere, whence on account of the general convergence theorem 2 . 2 . 2 , (proved in the following chapter), we may infer that almost everywhere 1 ^ lim -γ- ]£rk(x) = 0. Cantelli's theorem corresponds to the special case λη = η. The relation (22) is also equivalent to a theorem of BOREL, stating that the ratio of the number of the digits 1 to the number of the digits 0 among the first n digits in the dyadic representation of a number x between 0 and 1 tends to 1 for almost every x, when n —> o°.

Max (|a|, \b\)==A. We then obtain from (13), (14), (15) and (16) that ξ\δ \sn(ï)-m\^2AC^ Î-Ô f(t)-m t-ξ dt + 4s AC. FUNDAMENTAL IDEAS 29 Furthermore, by (12), we have i^m^h ζ-δ dt ξ-δ -ï\\\og\t-ï\ 2Κ (α—1)| log ô\a~x Since a—1 > 0 and d>0 may be chosen to be arbitrarily small, the quotient on the right-hand side can be made arbitrarily small, e. g.

O(x)qn(x) ==-. ΣCkPk(x)· k=n Now let M denote the maximum of σ(χ); then it follows from Schwarz's inequality (6) that 1 1 lç(x)o%x)ql(x)dxijç{x)pl(x)dx^ 4ü -1 1 -I ^ M \ç(x)o(x)ql{x)dx = M -1 and so a(xo)qn(xo) ^Z\ck\ k+n \Mxo)\ = (m+i)YMO(l) = 0(\). Thus we have established our intermediary theorem. Thus all the cases where a, ß are non-integer numbers are settled. In order to consider also integral values ay ß we use the fact (already mentioned) that for the Legendre polynomials (a = 0,ß = 0) the estimate p f °>(x) = 0(1) holds in (—1, 1).