The Typical Number Is a Lexicon
Cristian S. Calude, Tudor I. Zamfirescu · 1998
The parallelism between category and measure fails to hold true for random reals (i.e. reals having the sequence of digits in some base Chaitin-Martin-Lrandom, (7, 4)): constructively, they are measure-one sets of first category (12). In this note we strengthen some results in (6) by constructively proving that a class of pseudo-random reals has a -porous complement, so it is simultaneously residual and of measure-one. As a consequence, we constructively show that the typical number is a lexicon, i.e. even constructively most numbers do not obey any probability laws. To achieve our goal we prove a constructive version of (a weak form of) Lebesgue's Density Theorem, a result which might be interesting in itself.