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Topic: How WM is cheating - fat Cantor set measure
Replies: 3   Last Post: Jan 3, 2013 8:33 PM

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fom

Posts: 1,969
Registered: 12/4/12
Re: How WM is cheating - fat Cantor set measure
Posted: Jan 3, 2013 8:33 PM
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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.






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