On Dec 30, 12:17 am, fom <fomJ...@nyms.net> wrote: > ... > Unlike many other measures, Lebesgue > measure has an invariance property > that permits its product measures > to be defined without the general > theory of product measures. To > see why, consider the binary > expansions on the interval > > 0<=y<1 > > taking the eventually constant > sequences ending in constant 0 > as the representation for rational > numbers. ...
Not all rationals as binary expansions end with zeros, only multiples of inverse powers of two, for any finite string of zeros and ones there are expansions of rationals that end with those repeating.
"Thus, the claim that the non-zero measure of fat Cantor sets arises from relation with an atomic measurable space whose atoms are not singletons has been verified. "
Having measurable, countable, atoms, comprising units of measure: what would be their character as sets of reals?