Course in Probability Theory

110 1 LAW OF LARGE NUMBERS . RANDOM SERIESvk" ) (w) denote the number of digits among the first n digits of w that areequal to k . Then vk"~(w)/n is the relative frequency of the digit k in the firstn places, and the limit, if existing :lim v k" (60) = (Pk (w),n-*oo nmay be called the frequency of k in w . The number w is called simply normal(to the scale 10) iff this limit exists for each k and is equal to 1/10 . Intuitivelyall ten possibilities should be equally likely for each digit of a number picked"at random". On the other hand, one can write down "at random" any numberof numbers that are "abnormal" according to the definition given, such as• 1111 . . ., while it is a relatively difficult matter to name even one normalnumber in the sense of Exercise 5 below . It turns out that the number1 2345678910111213 . . .,which is obtained by writing down in succession all the natural numbers inthe decimal system, is a normal number to the scale 10 even in the stringentdefinition of Exercise 5 below, but the proof is not so easy . As fordetermining whether certain well-known numbers such as e - 2 or r - 3 arenormal, the problem seems beyond the reach of our present capability formathematics . In spite of these difficulties, Borel's theorem below asserts thatin a perfectly precise sense almost every number is normal . Furthermore, thisstriking proposition is merely a very particular case of Theorem 5 .1 .2 above .Theorem 5 .1 .3 . Except for a Borel set of measure zero, every number in[0, 1] is simply normal .PROOF . Consider the probability space (/, J3, m) in Example 2 ofSec . 2 .2 . Let Z be the subset of the form m/l0" for integers n > 1, m > 1,then m(Z) = 0 . If w E 'Z/\Z, then it has a unique decimal expansion ; if w E Z,it has two such expansions, but we agree to use the "terminating" one for thesake of definiteness . Thus we havew= •~ 1~2 . . .~n . . . .where for each n > 1, "() is a Borel measurable function of w . Just as inExample 4 of Sec . 3 .3, the sequence [~", n > 1) is a sequence of independentr.v .'s withfi n = k) _ ,lo , k=0,1, . . .,9 .Indeed according to Theorem 5 .1 .2 we need only verify that the ~"'s areuncorrelated, which is a very simple matter . For a fixed k we define the