Date: Dec 30, 2012 3:17 AM Author: fom Subject: How WM is cheating - fat Cantor set measure

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.