December 25, 2016 Sunday
Bedtime Story
Lebesgue Measure and Equidecomposability
Based on the Lebesgue measure, the Vitali Theorem can also be
stated as follows:
There exist subsets of R (real numbers) that are not Lebesgue
measurable.
Hence the Axiom of Choice needs to be invoked for dealing with
Vitali sets.
One of the many interesting properties that ensues from Lebesgue
measure is this:
If A is a disjoint union of countably many disjoint Lebesgue
measurable sets, then A is itself Lebesgue measurable and
(A) is equal to the sum (or infinite series) of the measures of
the involved measurable sets.
The implications of this is very counterintuitive.
It two open subsets of a line or a plane are equidecomposable then
they have equal area.
This means that any infinitely sized sets when split into subsets
will result in more infinitely sized subsets.
Let me give you an example of equidecomposability.
Take a unit circle A and another unit circle B with one missing
point Z.
These two circles are equidecomposable simply because the second
circle B can be made into a full complete circle.
Let me show you how.
In the circle B with the missing point Z make a subset E that consists
of all the points that are positive integer number of radians clockwise from
the missing point Z.
They are countable but infinite because of the irrationality of
pie.
Let all the rest all the numbers on the circle B be defined under
the subset F.
Then one pick up the subset E and rotate it anti clockwise by 1
radian.
The missing point at Z gets filled up by the point one radian to
the right of it, just like the point on the n-1 radian will get filled by the
point of the n radian.
Then put the subset E back with the subset F and you end up with a
full circle B that is same as the full circle A.
This proves the unit circle A and the unit circle B with a missing
point Z were and are equidecomposable.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
Felix Hausdorff had originally come with the concept of paradoxical decomposition of a set using the Axiom of Infinity.
Paradoxical decomposition is partitioning of a set into 2 subsets along with an appropriate group of functions on some universe, such that the partition can be mapped back onto the entire original set using only finitely many distinct functions.
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