Remarks on Bernstein sets
Dave L. Renfro
"Uncountable, separable (top. incomplete) space with no Cantor
set", Bernstein sets came up. About three years ago I spent
quite a bit of time writing up some notes about Bernstein sets,
Vitali sets, Hamel bases and additive functions, and various
results about decomposing the reals into a large collections
of large pairwise disjoint sets (or "almost" pairwise disjoint,
where "almost" means the pairwise overlaps are small in cardinality,
measure, Baire category, or some other notion). The notes are
still incomplete, and I haven't worked on them in a while (I've
been more interested in other things lately), but the beginning
parts that deal with Bernstein sets might be of interest anyway.
One day I hope to finish what I've started (for example,
in the time since then I've collected quite a few references
and results that belong below) and post everything in a
four or five part series on these topics, but since that
might not happen for a few years and what I've written
thus far on Bernstein sets might be of interest to others,
I'm posting this portion of my notes.
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I. NOTATION AND SOME PRELIMINARIES
\leq -- less than or equal to
R -- the set of real numbers
interval -- means an interval with positive length
perfect set -- means a nonempty perfect set
countable -- means finite or countably infinite
set difference -- A-B = {x in A: x does not belong to B}
m(E) -- the Lebesgue measure of the measurable set E
(m^*)(E) -- the outer Lebesgue measure of the set E
= inf { m(G): G is an open set containing P }
(m_*)(E) -- the inner Lebesgue measure of the set E
= sup { m(C): C is a closed set contained in E }
= m(I) - (m^*)(I-E) (when E belongs to a bounded interval I)
The first formulation of m_* is due to William H. Young and the
second is the definition that Lebesgue originally used. A bounded
set E is measurable if and only if (m^*)(E) = (m_*)(E), and an
arbitrary set is measurable if and only each of its intersections
with [-n,n] has inner measure equal to outer measure.
Both Young and Vitali obtained much of Lebesgue's integration
theory independent of Lebesgue (and of each other). The term
"Lebesgue integral" is due to Young, who began using it in
recognition of Lebesgue's priority and Lebesgue's more complete
development of the subject. [Hardy mentions this (without giving
the explanation I gave) near the bottom of p. 224 in his
biography of Young in J. London Math. Soc. 17 (1942). Also,
I've looked at many of Young's earlier papers and I've seen
this term gradually creep into them.]
Few real analysis texts say much about inner measure. For a
very thorough survey of relations involving set-theoretic
operations applied to inner and outer measure, see Shiffman [17].
Incidentally, Shiffman's Theorem 6 shows that [0,1] can be
expressed as a countable union of pairwise disjoint sets of
outer measure 1, using a Bernstein construction. However,
Shiffman's paper only contains one reference, to Vitali's 1905
paper where the existence of a non-measurable set is first
established.
The following is often useful.
Lemma: Let E and F be disjoint subsets of R. Then
(m_*)(E) + (m_*)(F) \leq (m_*)(E union F)
\leq (m_*)(E) + (m^*)(F) \leq (m^*)(E union F)
\leq (m^*)(E) + (m^*)(F).
meager -- any set that is a countable union of nowhere dense sets
non-meager -- a set that isn't meager
co-meager -- the complement of a meager set
Baire -- a set belonging to the sigma-algebra generated by
open sets and the first category sets
This is a Baire category analog of being measurable, since
measurable sets can be characterized as belonging to the
sigma-algebra generated by the open sets and the outer measure
zero sets. The usual term is "has the property of Baire" (or
"has the Baire property") -- I'm simply using a shorter name.
It is not difficult to show that a set is Baire if and only if
it has a meager symmetric difference with some open set. Every
Borel set (in fact, every analytic set) is both measurable and
Baire.
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II. BERNSTEIN SETS
Lemma: The following are equivalent for a subset P of any
interval I with left endpoint a and right endpoint b:
(a) (m^*)(P) = b-a
(b) (m_*)(I-P) = 0.
(c) For each subinterval J of I, (m^*)(P intersect J) = m(J).
(d) For each subinterval J of I, (m_*)(P intersect J) = 0.
(e) For each measurable subset E of I, (m^*)(P intersect E)
equals m(E).
(f) For each measurable subset E of I, (m_*)(I-P intersect E)
equals 0.
Definition: We say that a subset of R has "full outer measure"
if it has the above property relative to every bounded
open interval.
In Halmos [8] sets with full outer measure are called "thick".
A Baire category version of having maximal outer measure is to
have a non-meager intersection with every open interval, and a
Baire category version of containing no measurable subsets with
positive measure is to contain no non-meager Baire subset. The
following lemma says that these two properties are equivalent.
Note that this is a Baire category analogue of the equivalence
(c) <==> (f) from above. The proof is straightforward -- see
Morgan [12] (p. 73).
Lemma: The following are equivalent for a subset P of the reals.
(a) P has a non-meager intersection with every open interval.
(b) R-P contains no non-meager Baire subset.
Definition: We say that a set is "everywhere non-meager" if it
has the property in the lemma just given.
Paradoxically, a set and its complement can both have full outer
measure, and likewise with the property of being everywhere
non-meager.
Definition: We say that a set is "maximally non-measurable"
["maximally non-Baire"] if both the set and its
complement have full outer measure [are everywhere
non-meager].
The importance of these notions for us is that most of the
examples below are maximally non-measurable or maximally
non-Baire. This is because if we have a collection of two or
more pairwise disjoint sets with full outer measure, then each
of the sets in that collection must be maximally non-measurable.
Similarly for disjoint collections of everywhere non-meager sets.
Some other phrases that I've seen for this are "saturated
nonmeasurable" (Halperin [9]) and "m-thick" (Pettis [14]).
Sets P such that both P and the complement complement of P
have full outer measure are called M-shrets in Diamond/Gelles [7].
Definition: We say that B is a Bernstein set if B satisfies
any of the following equivalent contitions.
(a) B and its complement intersect every perfect set.
(b) B and its complement intersect every uncountable closed set.
(c) B and its complement intersect every uncountable Borel set.
(d) B contains no perfect subset.
(e) B contains no uncountable closed subset.
We can replace "Borel" with "analytic" in (c), since every
uncountable analytic set contains a perfect subset. I used
"Borel" because Borel sets are more familiar.
Note that the analog of (b) when "open set" is used in place of
"closed set" gives a dense set that has a dense complement, which
is quite a bit tamer!
Bernstein's 1908 [1] paper in which he originally proved the
existence of such a set dealt with trigonometric series uniqueness
issues. Bernstein gave his example in order to illustrate that
certain hypotheses in some of his results were necessary. This
is the Bernstein of the Cantor-Bernstein theorem about cardinal
inequality, but not the Bernstein who the Bernstein polynomials
are named after.
Bernstein sets can be constructed by a straightforward transfinite
induction "diagonal argument" on a well-ordering of the collection
of perfect sets. For the details, see Chapter 5 in Oxtoby [13] or
Gerald A. Edgar's April 21, 2000 sci.math post at
http://groups.google.com/group/sci.math/msg/a05611b50b4dc0b7
Incidentally, "nonempty intersection" in either (a) or (b) above
implies "intersection with cardinality c", since every
perfect set contains c-many pairwise disjoint perfect subsets.
For a short proof of this last fact that makes use of a
space-filling curve, see "LEMMA+" in my July 1, 2001 post at
http://groups.google.com/group/sci.math/msg/86ef3b7928eff8e9
Theorem: If B is a Bernstein set, then B and R-B both have full
outer measure and both are everywhere non-meager.
Proof: Let B be a Bernstein set and I be an open interval. If the
inner measure of "B intersect I" were positive, then this
intersection would contain a measurable set of positive
measure, hence a closed set of positive measure, hence a
perfect set of positive measure (Cantor-Bendixson), which
contradicts R-B having nonempty intersection with every
perfect set. Therefore, the inner measure of
"B intersect I" is zero. The same argument works for R-B.
Now suppose that "B intersect I" contains a non-meager
Baire subset. Then this intersection would contain a
perfect set
The phrase "totally imperfect" is often used to denote a set
which contains no perfect set. [The term "dispersed" also used,
especially in functional analysis.] Using this term it is immediate
that a set is a Bernstein set if and only if both it and its
complement are totally imperfect. Note that we can extract from the
previous proof the fact that any totally imperfect set has inner
measure zero and contains no non-meager Baire subset. [To see the
parallel, note that inner measure zero is equivalent to "contains
no non-zero-outer-measure measurable subset.] Thus, any totally
imperfect set is either SMALL (a meager measure zero set, and then
some) or BAD (a non-Baire, non-measurable set, and then some). In
particular, any Bernstein set is non-Baire, non-measurable, and
quite a bit worse.
The converse of the previous theorem is not true. Diamond/Gellès
[7] give a particularly simple example. Their example is
B union C, where B is any Bernstein set and C is any perfect
nowhere dense set with measure zero.
Curiously, although any Bernstein set in R is clearly totally
disconnected, every Bernstein set in R^n for n > 1 is connected!
Sierpinski [18] proved this in 1920. A proof can also be found
in Hausdorff [10] (p. 202). A minor modification of the Bernstein
construction (use nondegenerate closed connected sets in place of
perfect sets) is shown to satisfy a slightly stronger type of
connectivity in Hocking/Young [11] (p. 110) and Seebach/Steen [16]
(Example #124, p. 142).
The usefulness of Bernstein sets is often not realized. For instance,
let E be any measurable set with positive measure and B be any
Bernstein set. Then using the Lemma from above along with the fact
that B is full outer measure, it follows that "E intersect B" is a
non-measurable set having the same outer measure as E. This proves
the result for which the four page paper Pu [15] was devoted to.
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[1] Felix Bernstein, "Zur Theorie der trigonometrischen Reihen",
Sitzungsber. Sächs. Akad. Wiss. Leipzig. Math.-Natur.
Kl. 60 (1908), 325-338. [JFM 39.0474.02]
http://www.emis.de/cgi-bin/JFM-item?39.0474.02
[2] Jack B. Brown, "Density of one graph along another", Proceedings
of the American Mathematical Society 20 (1969), 147-149.
[MR 38 #1220; Zbl 165.06903]
http://www.emis.de/cgi-bin/Zarchive?an=0165.06903
[3] Jack B. Brown and Gregory V. Cox, "Classical theory of
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185-232. [MR 84g:54021; Zbl 503.54045]
http://www.emis.de/cgi-bin/MATH-item?0503.54045
[4] Celestyn Burstin, "Eigenschaften meßbarer und nichtmeßbarer
Mengen", Sitzungsberichte der Kaiserlichen Akademie der
Wissenschaften. Mathematisch-Naturwissenschaftlichen Klasse
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[5] Celestyn Burstin, "Die Spaltung des Kontinuums in $\kappa_1$
überall dichte Mengen", Sitzungsberichte der Kaiserlichen
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[JFM 45.0127.01]
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nichtmeßbare Mengen", Sitzungsberichte der Kaiserlichen
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[JFM 46.0293.02]
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[MR 11,504d; Zbl 40.16802]
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[9] Israel Halperin, "Non-measurable sets and the equation
f(x+y) = f(x) + f(y)", Proceedings of the American
Mathematical Society 2 (1951), 221-224.
[MR 12,685d; Zbl 43.11002]
http://www.emis.de/cgi-bin/Zarchive?an=0043.11002
[10] Felix Hausdorff, SET THEORY, 3'rd edition (translated from
the German by John R. Aumann, et al), Chelsea Publishing
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[MR 23 #A2857; Zbl 135.22701]
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Graduate Texts in Mathematics #2, Springer-Verlag, 1980,
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Proceedings of the American Mathematical Society 8 (1957),
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http://www.emis.de/cgi-bin/Zarchive?an=0079.32201
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Society 13 (1972), 267-270. [MR 46 #3730; Zbl 232.28002]
http://www.emis.de/cgi-bin/MATH-item?0232.28002
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IN TOPOLOGY, 2'nd edition, Dover Publications, 1978/1995,
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[18] Waclaw Sierpinski, "Sur les ensembles connexes et non
connexes", Fundamenta Mathematicae 2 (1921), 81-95.
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http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=2
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http://www.emis.de/cgi-bin/MATH-item?0732.54032
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Dave L. Renfro
Dave L. Renfro
> Definition: We say that B is a Bernstein set if B satisfies
> any of the following equivalent contitions.
>
> (a) B and its complement intersect every perfect set.
> (b) B and its complement intersect every uncountable closed set.
> (c) B and its complement intersect every uncountable Borel set.
> (d) B contains no perfect subset.
> (e) B contains no uncountable closed subset.
O-K, I warned everyone that these were the first part of
a rough draft! (d) and (e) should begin with "B is
uncountable and B contains ...".
Dave L. Renfro