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 <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

a_10, a_14

a_15

so, the differences between indexes proceeds

1,2,3,4,

2,3,4,

3,4,

4

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.

Thanks.