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Topic: How WM is cheating - fat Cantor set measure
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fom

Posts: 1,968
Registered: 12/4/12
How WM is cheating - fat Cantor set measure
Posted: Dec 30, 2012 3:17 AM
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To see that the measure associated with
fat Cantor sets can be attributed to
fintism in the final analysis, one
must look at the infinite product
of measure spaces.

Just as with the Tychonoff topology
on infinite product topologies,
infinite products of measure spaces
have a local finiteness constraint.
That is, since the measures of
measurable sets in the product will
be obtained by infinitary products,


m_oo(E)=Pi_n(m_n(E_n)) for n=0 to n=oo


the measure on the domain of each
component measure space must be
unity. That is, if

(X_i,S_i,m_i)


is a component measure space, then


m_i(X_i)=1


The local finiteness condition asserts
that measurable sets in the product
have measurable components satisfying


-(m_n(E_n)=1)


for only finitely many of its components.
a measureable set in the product. Thus,
for measurable sets in the product,


0<=m_oo(E)=Pi_n(m_n(E_n))<=1 for n=0 to n=oo


For present purposes, let


X_0={0,1}

S_0={{0,1}, {1}, {0}, null}

m_0({0,1})=1

m_0({1})=1/2

m_0({0})=1/2

m_0(null)=0


and, for i>0 let


(X_i,S_i,m_i)=(X_0,S_0,m_0)


and call the infinite product


(X_oo,S_oo,m_oo)


Now, each point of the Cartesian
product X_oo


X=(x_0,x_1,...)


is a countably infinite sequence of 1's
and 0's. Each finite initial segment
corresponds with a measurable set --
that is, a cylinder of points agreeing
on the first n coordinates. So, for an
initial segment of length n,


m_oo(x|n)= (1/2)^n


It should be clear that one obtains


m_oo(x)=lim_n(m_oo(x|n))=0 as n->0


so that denying a completed infinity
is equivalent to assigning non-zero
measures to the points of the product.



=================

To see that the fat Cantor set measure
is unrelated to the Lebesgue measure
on the interval 0<=p<1, observe first
that the measure on the set of points
for which only finitely many of the
coordinates is different from one, is
zero.

That is, for each such point, m_oo(x)=0.
There are only countably many finite
subsets and measures are countably
additive set functions. So, one
can form the set consisting of the
constant sequence,

x=(1,1,1,...)

and those sequences that differ from
x at finitely many indices. This set
has m_oo=0.

Next, take the points in the complement
of this set. Identify each of these
points with the sum of coordinates,

p(x)=Sigma_n(x_n/2^(n+1)) for n=0 to n=oo

and make the usual identification between
eventually constant sequences corresponding
to rational numbers.

Then, for each A such that

A={p(x)|0<=a<=p(x)<b<=1}

A is Lebesgue measurable and the Lebesgue
measure is

m_L(A)=(b-a)



=================

To see that fat Cantor set measure,
with respect to foundational considerations
arises from finitism, one must look
at the structure of Lebesgue measurable
sets in relation to the Borel
hierarchy.

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. Let



y=(y_0, y_1, y_2, y_3, y_4, y_5, y_6, y_7, y_8, y_9, y_10, y_11, ...)



be one such number. If one now
writes



x_1=(y_0, y_3, y_6, y_9, ...)

x_2=(y_1, y_4, y_7, y_10, ...)

x_3=(y_2, y_5, y_8, y_11, ...)



one obtains either three eventually
constant sequences or three sequences
that never become eventually constant.
Moreover, by the invariance of the
Lesbegue measure in relation to this
process, the transformation of a
Lebesgue measurable set yields three
Lebesgue measurable sets having the
same measure under the product.

The transformation to an infinitary
product is done with the usual
diagonal strategy,


x_1=(y_0, y_1, y_5, y_6, y_14, ...)

x_2=(y_2, y_4, y_7, y_13, ...)

x_3=(y_3, y_8, y_12, ...)

x_4=(y_9, y_11, ...)

x_5=(y_10, ...)

x_6=(...)


so that each Lebesgue measurable set
corresponds to an infinitely countable
collection of Lebesgue measurable sets
whose union has the same measure as
the original.

Now, the measurable spaces over
which Lebesgue measures are defined
are those spaces whose sigma-algebra
is generated by the open sets of the
topology and whose atoms correspond
with singletons. That is, Lebesgue
measures are defined with respect
to the Borel hierarchy.

Each Lebesgue measurable set differs
from a Borel set by a set of measure
zero. So, in fact, it is the Borel
sets that are responsible for the
invariance demonstrated by the
Lebesgue measure in this transformation.

But, one does not capture the
individuation of the Borel
sigma-algebra without invoking
completed infinities. Consequently,
as is seen with the non-zero measure
of the fat Cantor sets, "set of measure
zero" has no meaning without the
completed infinity invoked at each
stage of definition for the Borel
hierarchy.

Thus, the claim that the non-zero
measure of fat Cantor sets arises
from relation with an atomic
measurable space whose atoms are
not singletons has been verified.





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