Problem 22
Borel and A Different Kind of Normality (1909)
Problem. A real number in [0,1] is said to be normal1 in a given base if any finite pattern of digits occurs with the same expected frequency when the number is expanded in that particular base. Thus a number in [0,1] is normal in base 10 if, in its decimal expansion, each of the digits {0,1, . . ., 9} occurs with frequency 1/10, each of the pairs {00, 01, . . ., 99} occurs with frequency 1/100, and so on. Prove that almost every real number in [0,1] is normal for all bases.
Solution. Let the random variable 
(
= 0, 1, . . ., 9; j = 1, 2, . . .) be the jth digit in the decimal expansion of 
, that is
Then, for the digit b = 0, 1, . . ., 9, we define the binary random variable
It is reasonable to assume that the 
s are independent (across the j 's)2 and identically distributed. Applying the Strong Law of Large Numbers (SLLN),3 we ...
