On 12/31/2012 10:16 AM, Ross A. Finlayson wrote: > 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.
On looking closer, I should have written the infinite case like
a_1, a_2, a_4, a_7, a_11
a_3, a_5, a_8, a_12
a_6, a_9, a_13
so, the differences between indexes proceeds
Thus every rational number maps to rational numbers either through becoming eventually constant or by having a repeating sequence that recurs in each derived sequence relative to modulo arithmetic.