Vitali nonmeasurable
Alexander Abian
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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Robert Israel
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
Alexander Abian
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
Alexander Abian
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].