L.E.D. (and Cantor set)

[Ralph Howard]

Andrew writes:

> 
> Stephen Deng writes:
> > Maybe the emptiness aspect has something to do with properties of Cantor 
> > sets, which are fractals.  The Cantor set is the set of points on the 
> > interval [0,1] where you first cut out the middle third.  In the next 
> > iteration, cut out the middle third of the remaining two pieces and so on 
> > for an infinite number of iterations.  The resulting set has dimension 2/3, 
> > but it has a probabilistic measure of 0 which means that if you throw a 
> > dart at the interval [0,1] (and you are a good enough dart player to his a 
> > straight line), there is a 0% probability (in measure theoretic terms) that 
> > you will hit one of the points in the set.  Therefore, in essence there is 
> > something there which is mostly emptiness.
> 
> I believe this is usually called Cantor's Ternary set. And you are
> right that measure theory probably provides a better basis for a
> redefinition of `empty' than anything based on the space we inhabit,
> particularly a naive assimilation with 3-d Euclidean space. Measure
> theory provides a conservative extension of the notion of `extent' or
> `length' (mathematicians call it `measure').
> 
> But note that the set you are describing is very peculiar. It is
> *discontinuous* everywhere on [0, 1] i.e. take any point which lies in
> the set e.g. the point X, 1/3rd the way along the interval (i.e. the
> one with coordinate 0.3333...). If you consider any subinterval [X -
> eps, X + eps], where eps is a number between 0 and 1, you will find
> that the interval contains points which are in the Cantor Ternary set
> and points which are not. Equivalently take any subinterval [a, b]
> where 0 < a, b < 1 and the interval will contain both points inside
> and points outside the Ternary set no matter how small the difference
> between a and b. Clearly the set `extends' through the interval [0, 1]
> in a very different way to a conventional interval such as [0.2, 0.4]).

First, and I realize it makes no difference to the rest of what is
being said here, the dimension of the Cantor set is ln(2)/ln(3) and
not 2/3.  As for the statement that the Cantor set `extends' through
out the interval [0 ,1] I may be misunderstanding, but this does not
seem to be true.  Stephen's description of the set by repeatedly
removing open intervals is exactly correct the Cantor set what is left
over after these deletions.  Thus the deleted intervals and any of
their subintervals contain no points of the Cantor set.  For example
the interval [.4, .6] contains no points of the set.  It is however
the case that every interval, no matter how small, will contain points
not in the Cantor set.

> 
> Also, when you say that the set has dimension 2/3 but has
> probabilistic measure of 0 you are masking something in the way you
> phrase things. What you are actually referring to here is that by
> employing one measure you can arrive at an `extent' or `length' for
> the set of 2/3 whereas using another measure it has `extent' 0. Which
> one is its real length? Only legitimate answer is `neither' or `it
> depends'.
> 
> Further, the pecularities are all down to the peculiar construction of
> the set, not anything wierd in the original notion of measure. For the
> usual measures employed to prove this result the interesting thing is
> that both measures agree on conventional intervals, they only disagree
> on sets which have discontinuities like the Ternary set. So, no
> mystery, nothing really surprising, nothing in the result that was not
> carefully placed there by adopting the appropriate assumptions and
> definitions. That's all this talk of dart playing reduces to, really.
> A parlour trick where you substitute the lady for a joker half-way
> through.
> 

It is a very good point that exactly what measure is being used must
be specified and not changed.  However I don't think that the
peculiarities can just be passed off on the Cantor middle third set.
By doing a variant of the same construction, but taking a different
proportion out each step, for example by talking out middle fifths (or
middle one hundreds or by even changing the proportions at each step)
we can get sets (still called Cantor sets) that have any dimension
between from 0 to 1 and there are even Cantor having positive measure
as close to one as we like say .99.  Thus there is such a set so that
if we do Stephen's dart experiment then we will hit the set 99 times
out of a hundred (using the standard "length" measure on [0,1]).

What makes things look paradoxical is that much of modern probability
is done on a spaces, like the unit interval, where any single outcome
of the "experiment" has probability zero.  For example if the expement
is choosing a number at random form the interval (i.e. throwing a dart
at this interval) the probability of hitting .5 exactly is zero (while
hitting between .45 and .55 has probability 1/10).  More surprising is
that some infinite sets (the rational numbers, the middle thirds set)
have probability zero.  It was well into this century before
mathematicians where to set up a model of probability so that the
statements:

1. The probability that anyone has any particular number as a height
   is zero

2.  Everyone does have a height

don't contradict each other.  Putting it this way seems to me to
emphasize that it is the nature of the measures used in probability at
least as much as the construction of any particular set that leads to
the set being considered negligible.

I have not gotten far enough into M&D to know if this is relevant to
Stephen's original remarks.

Sorry for pedantic and going on so long on this, but it is a problem I
am fond of.

Ralph

-- 
Ralph Howard                    Phone:  (803) 777-2913 
Department of Mathematics       Fax:    (803) 777-3783 
University of South Carolina    e-mail: howard at math.sc.edu
Columbia, SC 29208 USA          http://www.math.sc.edu/~howard/