fom
Posts:
1,968
Registered:
12/4/12


How WM is cheating  fat Cantor set measure
Posted:
Dec 30, 2012 3:17 AM


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(xn)= (1/2)^n
It should be clear that one obtains
m_oo(x)=lim_n(m_oo(xn))=0 as n>0
so that denying a completed infinity is equivalent to assigning nonzero 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)=(ba)
=================
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 sigmaalgebra 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 sigmaalgebra without invoking completed infinities. Consequently, as is seen with the nonzero 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 nonzero measure of fat Cantor sets arises from relation with an atomic measurable space whose atoms are not singletons has been verified.

