Vitali nonmeasurable

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Vitali nonmeasurable Alexander Abian 1/9/97 12:00 AM

Referring to the classical example of Vitali's Lebesgue nonmeasurable
subset  V of the real unit interval - is it the case that the outer
measure of any such  V  is equal to  1 ?

Please answer at your earliest convenience.  Thank you.
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Vitali nonmeasurable Robert Israel 1/9/97 12:00 AM

In article <abian.8...@class1.iastate.edu>, ab...@iastate.edu (Alexander Abian) writes:
 
|> Referring to the classical example of Vitali's Lebesgue nonmeasurable
|> subset  V of the real unit interval - is it the case that the outer
|> measure of any such  V  is equal to  1 ?

Do you mean a set V that contains one representative of each coset x+Q where Q is
the rationals?  No, you can always take these representatives to be in any
given interval [a,b] with 0 <= a < b <= 1, so the outer measure of V is at most
b-a.  

What's a bit more challenging, I think, is to find a V whose outer measure is equal to 1.  You can do it as follows:

The family F of open subsets of [0,1] with measure < 1 has cardinality c.
So it can be well-ordered in such a way that every member of F has fewer than
c predecessors.  Since the complement of each member of F has cardinality
c, there is a function f: F -> [0,1] such that for each A in F, f(A) is not a member of A and is not in {f(B)+q: B a predecessor of A, q in Q}.  Thus
{ f(A): A in F } contains at most one representative of each coset.  Complete
V by putting in representatives of all cosets not already represented.
Then the outer measure of this V is 1.

Robert Israel                            isr...@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Y4
 

Vitali nonmeasurable Alexander Abian 1/9/97 12:00 AM


Referring to the classical example of Vitali's Lebesgue nonmeasurable
subset  V of the real unit interval - is it the case that the outer
measure of SOME such  V  is equal to  1 ?

Please answer at your earliest convenience.  Thank you.


PS. this is a correction of an earlier posting where  instead of  SOME
    "any"  was typed by mistake

--

--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)

Vitali nonmeasurable Alexander Abian 1/10/97 12:00 AM


In article <5b3i2a$5u4$1...@nntp.ucs.ubc.ca>,


Robert Israel <isr...@math.ubc.ca> wrote:
>In article <abian.8...@class1.iastate.edu>, ab...@iastate.edu (Alexander Abian) writes:

[the word  "SOME"  is added  by Abian (as explained by underlined sentence)


>> Referring to the classical example of Vitali's Lebesgue nonmeasurable
>> subset  V of the real unit interval - is it the case that the outer
>> measure of SOME    (instead of mistakingly originally typed "any") such
>> V is equal to  1 ? ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
Abian answers:
 
Dear Mr. Israel,

 Probably you did not notice that very soon after my posting that you
have quoted above, I posted another one (which is still posted) by
correcting the typo which I indicated in your quote above.
..............................................................

Isreal continues:

>................you can always take these representatives to be in any


>given interval [a,b] with 0 <= a < b <= 1, so the outer measure of V is
>at most b-a.  
>What's a bit more challenging, I think, is to find a V whose outer measure is
>equal to 1.  You can do it as follows:

>The family F of open subsets of [0,1] with measure < 1 has cardinality c.
>So it can be well-ordered in such a way that every member of F has fewer than
>c predecessors.  Since the complement of each member of F has cardinality
>c, there is a function f: F -> [0,1] such that for each A in F, f(A) is not
>a member of A and is not in {f(B)+q: B a predecessor of A, q in Q}.  Thus
>{ f(A): A in F } contains at most one representative of each coset.  Complete
>V by putting in representatives of all cosets not already represented.
>Then the outer measure of this V is 1.
>
   Abian answers:

  Your answer seems  correct.

  The reason for my  posting the question was that I had to present an example
of V  with outermeasure 1 and I wanted to see if there is an example
simpler than the one I have devised.
  My devised  example  is based on the fact that " If a subset S  of  [0,1]
has a nonempty intersection with every closed subset of positive (Lebesgue)
measure of  [0,1]  then  the  outermeasure of S  is equal to  1.
  This result can be found in my paper
     
          "A SIMPLEST EXAMPLE OF A NONMEASURABLLE SET"

 Simon Stevin Math Journal, September 1976 (vol 2?) pp. 101-102

  Based on the above, I recently gave a construction of  V  with  outer measure
  equal  1.

 Moreover, based on a variant of the same result,  I recently gave examples
of  V  with outer measure  precisely equal to  b - a  and not only "at most"
for  a  an  b  as mentioned by you.  All one has to do is to consider
the set of all  closed subsets of [a,b] which have positive  Lebesgue)
measure,  then   pick up a point from each of them to include in the
construction of a  V  by appropriate addition of appropriate points not
used from  [a, b].

--

--------------------------------------------------------------------------
   ABIAN MASS-TIME EQUIVALENCE FORMULA  m = Mo(1-exp(T/(kT-Mo))) Abian units.
       ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS  AND EPIDEMICS
       ALTER THE SOLAR SYSTEM.  REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT  
                     TO CREATE A BORN AGAIN EARTH (1990)