fom
Posts:
1,031
Registered:
12/4/12
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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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