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In measure theory, a non-measurable set is one
which does not belong to the algebra of measurable sets of some measurable space. The usual context for this is stated in terms of translation invariant measures on the
set of reals R.
Theorem. There is no countably additive measure on all subsets of R which is translation
invariant and is finite and non vanishing on [0,1].
This follows from Theorem D, Section 16, of the Halmos reference below. That theorem requires use of the axiom of choice. It
follows immediately from this theorem, that non-measurable sets exist for any countably additive translation invariant measure
which is finite and non vanishing on [0,1]. This is true in particular, for the algebra of Caratheodory-measurable subsets of
R relative to the outer measure defined as follows:
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where the infimum above is taken over all countable covers of A by intervals
{Ii}i.
Non-measurable sets are highly non-constructive and mathematicians consider them to be extremely pathological.
A slight extension of the above theorem is as follows:
Theorem. There is no countably additive measure on all subsets of Rn which is
translation invariant and is finite and non vanishing on the n-dimensional cube [0,1]n.
The decomposition of the unit ball in R3 into five disjoint subsets as stated in the Banach-Tarski paradox involves non-measurable sets.
For an explicit "construction" of a non-measurable set, see Vitali set.
References
- Paul R. Halmos, Measure Theory, D. van Nostrand Co., 1950.
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