Date: Jan 3, 2013 8:33 PM
Author: fom
Subject: Re: How WM is cheating - fat Cantor set measure

On 12/31/2012 10:16 AM, Ross A. Finlayson wrote:
> On Dec 30, 12:17 am, fom <> 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

a_10, a_14


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.