Re: intersection of measurable sets
Toni Lassila
wrote:
>Let {E_k}^{k=1}^\infty be a sequence of measurable subsets in [0,1]
>such that m(E_k) = 1 for all k. Show m(\bigcap_{k=1}^\infty E_k) = 1.
Clearly the intersection if measurable (why?). Assume it has measure
less than 1. What can you say about the sequence of finite
intersections, A_n := \bigcap_{k=1}^\infty E_k? Does the measure of
these sets converge somewhere? Does this help you find a pair of sets
E_k with an interesting property? This should lead to a contradiction.
Dave L. Renfro
> Let {E_k}^{k=1}^\infty be a sequence of measurable
> subsets in [0,1] such that m(E_k) = 1 for all k.
> Show m(\bigcap_{k=1}^\infty E_k) = 1.
Can you show the measure of its complement relative
to [0,1] is zero? Hint: De Morgan.
Here's something with more teeth to it:
Let E_1, E_2, ... be an infinite sequence of subsets
of [0,1] and C be a fixed positive real number. Assume
that the measure of E_n is greater than C for each n.
Does there exist a subsequence {E_n_k} of {E_n} such
that the intersection of all the E_n_k's is nonempty?
What if we replace "is nonempty" with "has positive
measure"?
Dave L. Renfro
The World Wide Wade
<1115855306....@z14g2000cwz.googlegroups.com>,
"abalone" <rooro...@gmail.com> wrote:
> I was thinking using the fact that each E_k is measurable, so we can
> find a closed set F_k in E_k with property that m(E_k - F_k) < e/2^n.
> And then the m([0,1] - \bigcap F_k) = m(\bigcup ([0,1] - F_k) <= \sum
> m([0,1] - F_k) = e
These closed sets are not helping; try dealing directly with the
E_k's.
David C. Ullrich
wrote:
>abalone wrote:
>
>> Let {E_k}^{k=1}^\infty be a sequence of measurable
>> subsets in [0,1] such that m(E_k) = 1 for all k.
>> Show m(\bigcap_{k=1}^\infty E_k) = 1.
>
>Can you show the measure of its complement relative
>to [0,1] is zero? Hint: De Morgan.
>
>Here's something with more teeth to it:
>
>Let E_1, E_2, ... be an infinite sequence of subsets
>of [0,1] and C be a fixed positive real number. Assume
>that the measure of E_n is greater than C for each n.
>Does there exist a subsequence {E_n_k} of {E_n} such
>that the intersection of all the E_n_k's is nonempty?
Twice just now I was about to post a wrong yes/no answer
and caught myself just in time. So I think this counts
as hard enough that we should give a hint:
First, we can assume that each A_n is closed (why?).
Now, for each n, either
(i) m(E_n intersect [0,1/2]) > C/2
or
(ii) m(E_n intersect [1/2,1]) > C/2
(why?). Hence either (i) holds for infinitely many
n or (ii) holds for infinitely many n. (If you want
at this point you could simplify the notation by
assuming that (i) holds for infinitely many n.)
Now...
>What if we replace "is nonempty" with "has positive
>measure"?
This is very interesting, a measure-theoretic thing
where it's possible to get a null set but not possible
to get an empty set. I guess there must be uncountably
many increasing sequences (n_k)...
>Dave L. Renfro
************************
David C. Ullrich
Dave L. Renfro
> Here's something with more teeth to it:
>
> Let E_1, E_2, ... be an infinite sequence of subsets
> of [0,1] and C be a fixed positive real number. Assume
> that the measure of E_n is greater than C for each n.
> Does there exist a subsequence {E_n_k} of {E_n} such
> that the intersection of all the E_n_k's is nonempty?
> What if we replace "is nonempty" with "has positive
> measure"?
The weaker statement can be proved using a set
version of Fatou's Lemma (apply Fatou's lemma to
characteristic functions). Let m(B) denote the
measure of a set B, "leq" be "less than or equal
to", and "geq" be "greater than or equal to".
Then, keeping in mind that all the measures are
bounded, we have
m(lim-inf E_n) leq lim-inf[m(E_n)]
leq lim-sup[m(E_n)] leq m(lim-sup E_n).
Hence, m(lim-sup E_n) is at least C. Thus, the set of
points that belong to infinitely many of the E_n's has
measure at least C. In particular, there exists at least
one point belonging to infinitely many of the E_n's,
and this point gives rise to a subsequence of {E_n}
that has nonempty intersection.
You'd think there would be enough maneuvering room in
the above proof to get a subsequence having a positive
measure intersection, but this is not possible in
general. The problem is that although "measure at
least C" tells us there are uncountably many, and
then some, points belonging to infinitely many of
the E_n's, the fact that the collection of "infinitely
many E_n's" can vary from point to point prevents us
from easily getting a handle on how large the intersection
is for a fixed subsequence of {E_n}.
As a matter of fact, the positive measure intersection
is false in general. Here's an example from
Chernoff/Waterhouse [3]:
Let E_n be the set of points in [0,1] whose b-ary
expansion is different from 0 for its n'th digit.
Then m(E_n) = 1 - 1/b for each n and the intersection
of any k-many of the E_n's has measure (1 - 1/b)^k,
which implies that the intersection of every infinite
subcollection of the E_n's has measure zero.
Essentially the same example is in Halmos [9]
(Problem 14G, pp. 124-125, 316), Halmos [10] (section
on Rademacher sets), and Benedetto [1] (Problem 2.8(a),
p. 66).
Note this shows that no matter how large 'C' is,
as long as 'C' is less than 1 we cannot get a subsequence
whose intersection has positive measure. However, if
lim-sup[m(E_n)] = 1, then there will exist a subsequence
whose intersection has positive measure. In fact, the
measure of the intersection of some subsequence can be
made arbitrarily close to 1. This result can be found
in Benedetto [1] (Problem 2.8(b), p. 66), in
Chernoff/Waterhouse [3], and in [8] (Problem 1.6 of
Chapter VIII, p. 87; solution sketch on p. 263).
Problem 1.6 in [8] also asks if the measure of the
intersection of some subsequence can be positive
if lim-inf[m(E_n)] > 3/4.
Proof: choose a subsequence of {E_n} such that the
measures of the sets in this subsequence approach 1,
and then thin out this subsequence sufficiently to
produce a subsequence {E_n_k} such that the sum from
k=1 to infinity of 1 - m(E_n_k) is less than 1. Then
we have
1 > sum(k=1 to inf) of m( [0,1] - E_n_k )
geq m[ union(k=1 to inf) of [0,1] - E_n_k ]
= m[ [0,1] - intersection(k=1 to inf) of E_n_k ].
If we additionally thin out the subsequence so that
the sum from k=1 to infinity of 1 - m(E_n_k) is less
than any preassigned positive number, then we can
arrange for the measure of the intersection to be
arbitrarily close to 1.
Although we've seen that it's possible for there to be
no subsequence whose intersection has positive measure,
Erdos/Kestelman/Rogers [4] proved that we can always
find a subsequence whose intersection has cardinality c.
In fact, they proved a result slightly stronger than
the following:
There exists a subsequence and a Borel set B with
m(B) geq C such that every point of B is a condensation
point of the intersection of this subsequence.
Moreover, Erdos/Taylor [5] proved:
Given any Hausdorff measure function h such that the
Hausdorff h-measure of an interval is infinite, there
exists a subsequence whose intersection has infinite
Hausdorff h-measure. In particular, we can find a
subsequence whose intersection has Hausdorff dimension 1.
Finally, it is shown in Scoville [17] that we can find
a subsequence {E_n_k} with nonempty intersection such
that lim-sup(k --> inf) of k / n_k is geq C (i.e. the
subsequence has upper density C), and this is the best
possible result in the sense that we can have
lim-sup(k --> inf) of k / n_k leq inf{m(E_n): n = 1, 2, ...}
for every subsequence with a nonempty intersection.
[The first half of Scoville's proof does not appear to
be correct. However, the problem was originally posed
by Erdos, who apparently submitted a solution along with
his proposal since the problem is not marked with a * to
indicate the proposer didn't supply a solution. Thus,
I strongly suspect the statement itself is true.]
Regarding uncountable collections, what remains will
be a sketch of an interesting proof in Gillis [7].
For some perspective, note that if S is a collection of
pairwise disjoint subsets of R with positive measure,
then S is countable.
This can fail badly for non-measurable sets, by the way.
Lusin/Sierpinski [14] proved there exists a collection
of c-many pairwise disjoint subsets of [0,1] each having
outer measure 1.
Theorem (Gillis): Let 0 < C < 1 and assume that S
is a collection of measurable subsets of [0,1]
such that S is uncountable and m(E) geq C for each
E in S. Then for each epsilon > 0 we can find
subcollections S" and S' of S with S" contained
in S' and S' contained in S such that:
(a) S' is uncountable and, for each E,F in S', we have
m(E intersect F) > C - epsilon.
(b) S" is countably infinite and the measure of
the intersection of S" is greater than C - epsilon.
Note: Halmos [9] (solution to problem 14H, pp. 316-317)
and Halmos [10] (p. 308) give a proof of (a), but there
is no mention of Gillis' paper.
STEP 1: Let M[0,1] be the collection of measurable
subsets of [0,1] modulo the equivalence relation
E eq. F <==> m(E symm. diff. F) = 0. Define a map
d: M[0,1] x M[0,1] --> [0, inf) by d(E,F) = measure
of the symmetric difference of E and F. Then d
defines a metric on M[0,1].
Note: This metric space has a number of interesting uses.
The text Bruckner/Bruckner/Thomson [2] gives many of
them -- Items 8.22 (p. 357), 9.12 (p. 378), 9:3.4
and 9:3.8 (p. 387), 9.34 (p. 392), 9:16.3 (pp. 430-431),
10:1.12 (p. 440), 10.17 (pp. 455-456), 13.8 (pp. 580-582),
and 13:5.2 (p. 587). The result in Exercise 10:1.12 is
particularly interesting: "In the Baire category sense,
almost all E in M[0,1] have the property that both E and
[0,1] - E intersect every open interval in [0,1] in a
set of positive measure." This result is also asked for
in Kharazishvili [12] (Exercise 2, pp. 78-79). A proof
can be found in Kirk [13], buried within the proof of
his Theorem 1 (p. 886). A related interesting result is
given in Kato/Kanzo/Shinnosuke [11]. They consider the
space that I'll call N[0,1], which consists of *all* subsets
of [0,1] modulo the same equivalence relation, where the
distance is defined in the same way except that outer
Lebesgue measure is used. They prove that M[0,1] is a
perfect nowhere dense set in N[0,1]. In other words, the
collection of measurable subsets of [0,1] forms a very
tiny part of the collection of all the subsets of [0,1]!
STEP 2: (M[0,1], d) is a separable metric space.
This can be proved by showing that the collection of
all finite unions of open subintervals of [0,1] with
rational endpoints is a dense subset of M[0,1].
ADDITIONAL FACT: M[0,1] is a complete metric space.
This isn't needed for the proof of Gillis' theorem,
but it is needed so that Baire category result I
mentioned just above has significance. One way to
prove completeness is to note that d(E,F) is equal
to the integral over [0,1] of the absolute value
of the difference of the characteristic functions
of E and of F, and then make use of theorems involving
Lebesgue integration. For example, it would be
enough to prove that M[0,1] is a closed subset of
L^1[0,1], since L^1[0,1] is complete. (This is the
approach Zaanen [20] uses on pp. 80-89.) For a direct
proof that doesn't rely on Lebesgue integration,
see Oxtoby [15] (p. 44), Rao [16] (pp. 87-89),
or Taylor [19] (the extensive hint for problem #13
in Section 4-8, pp. 214-215).
STEP 3: Every uncountable subset of a separable metric
space contains a condensation point.
STEP 4: Since S is an uncountable subset of M[0,1],
it follows that S contains a condensation point
E*. To prove (a), let S' be the collection
of all sets E in M[0,1] such that
d(E,E*) < epsilon/2. Then S' is uncountable
and, for each E,F in S', we have d(E,F) < epsilon.
In particular, m( E - (E intersect F) ) is
less than epsilon, and hence the measure of
E intersect F is greater than C - epsilon.
STEP 5: To prove (b), choose E_n in S' for each positive
integer n so that d(E_n, E*) is less than
epsilon / 2^n. Let A_n = E* - (E_n intersect E*)
and A be the union of the A_n's. Then for each n,
m(A_n) < epsilon / 2^n and m(A) < epsilon.
Moreover, for each n we have E* - A contained
in E* - A_n contained in E_n, and thus E* - A
is contained in the intersection of the E_n's.
Therefore, the measure of the intersection of
the E_n's is greater than C - epsilon, and hence
we can take S" = {E_n in S': n = 1, 2, ...}.
[1] John J. Benedetto, REAL VARIABLE AND INTEGRATION,
Mathematische Leitfäden. Stuttgart: B. G. Teubne, 1976.
[MR 58 #28328; Zbl 336.26001]
http://www.emis.de/cgi-bin/MATH-item?0336.26001
[2] Andrew M. Bruckner, Judith B. Bruckner, and Brian S.
Thomson, REAL ANALYSIS, Prentice-Hall, 1997.
[Zbl 872.26001]
http://www.emis.de/cgi-bin/MATH-item?0872.26001
[3] Paul Robert Chernoff and William C. Waterhouse, "Solution
to Monthly Problem 5408", American Mathematical Monthly
74 (1967), 743-744.
[4] Paul Erdos, Hyman Kestelman, and Claude Ambrose Rogers,
"An intersection property of sets with positive measure",
Colloquium Mathematicum 11 (1963), 75-80.
[MR 28 #2182; Zbl 122.29903] [available on-line]
http://www.emis.de/cgi-bin/MATH-item?0122.29903
http://matwbn.icm.edu.pl/tresc.php?wyd=8&tom=11
[5] Paul Erdos and Samuel James Taylor, "The Hausdorff measure
of sets of positive Lebesgue measure", Mathematika 10
(1963), 1-9. [MR 27 #3765; Zbl 141.05501]
http://www.emis.de/cgi-bin/MATH-item?0141.05501
[6] Joseph E. Gillis, "Note on a property of measurable
sets", Journal of the London Mathematical Society
11 (1936), 139-141. [Zbl 14.05501; JFM 62.0243.04]
http://www.emis.de/cgi-bin/Zarchive?an=0014.05501
http://www.emis.de/cgi-bin/JFM-item?62.0243.04
[7] Joseph E. Gillis, "Some combinatorial properties of
measurable sets", Quarterly Journal of Mathematics
(Oxford) 7 (1936), 191-198. [Zbl 14.39504; JFM 62.0244.01]
http://www.emis.de/cgi-bin/Zarchive?an=0014.39504
http://www.emis.de/cgi-bin/JFM-item?62.0244.01
[8] M. G. Goluzina, A. A. Lodkin, B. M. Makarov, and A. N.
Podkorytov, SELECTED PROBLEMS IN REAL ANALYSIS,
Translations of Mathematical Monographs #107, The
American Mathematical Society, 1992.
[MR 93i:26001; Zbl 765.26001]
http://www.emis.de/cgi-bin/MATH-item?0765.26001
[9] Paul R. Halmos, PROBLEMS FOR MATHEMATICIANS YOUNG AND
OLD, Dolciani Mathematical Expositions #12, The
Mathematical Association of America, 1991.
[MR 92j:00009; Zbl 814.00004]
http://www.emis.de/cgi-bin/MATH-item?0814.00004
[10] Paul R. Halmos, "Large intersections of large sets",
American Mathematical Monthly 99 (1992), 307-312.
[MR 93c:28003; Zbl 759.28001]
http://www.emis.de/cgi-bin/MATH-item?0759.28001
[11] Nobuyuki Kato, Tadashi Kanzo, and Oharu Shinnosuke,
"A note on the measure problem", International Journal
of Mathematical Education in Science and Technology
19 (1988), 315-318. [Zbl 637.28002]
http://www.emis.de/cgi-bin/MATH-item?0637.28002
[12] Alexander B. Kharazishvili, STRANGE FUNCTIONS IN REAL
ANALYSIS, Marcel Dekker, 2000.
[MR 2001h:26001; Zbl 942.26001]
http://www.emis.de/cgi-bin/MATH-item?0942.26001
[13] Ronald Brian Kirk, "Sets which split families of
measurable sets", American Mathematical Monthly 79
(1972), 884-886. [MR 47 #5201; Zbl 249.28003]
http://www.emis.de/cgi-bin/MATH-item?0249.28003
[14] Nikolai N. Lusin and Waclaw Sierpinski, "Sur une
décomposition d'un intervalle en une infinité non
dénombrable d'ensembles non mesurables" [On a
decomposition of an interval into a nondenumerably
many nonmeasurable sets], Comptes Rendus Académie
des Sciences (Paris) 165 (1917), 422-424.
[JFM 46.0294.01] [available on-line]
http://www.emis.de/cgi-bin/JFM-item?46.0294.01
[15] John C. Oxtoby, MEASURE AND CATEGORY, 2'nd edition,
Graduate Texts in Mathematics #2, Springer-Verlag, 1980.
[MR 81j:28003; Zbl 435.28011]
http://www.emis.de/cgi-bin/MATH-item?0435.28011
[16] Malempati M. Rao, MEASURE THEORY AND INTEGRATION,
2'nd edition, Monographs and Textbooks in Pure and
Applied Mathematics #265, Marcel Dekker, 2004.
[MR 2004k:28003]
[17] Richard Scoville, "Solution to Monthly Problem 5074",
American Mathematical Monthly 71 (1964), 105.
[18] Karl R. Stromberg, INTRODUCTION TO CLASSICAL REAL
ANALYSIS, Wadsworth International, 1981.
[MR 82c:26002; Zbl 454.26001]
http://www.emis.de/cgi-bin/MATH-item?0454.26001
[19] Angus E. Taylor, GENERAL THEORY OF FUNCTIONS AND
INTEGRATION, Blaisdell/Dover, 1965/1985.
[MR 87b:26001; Zbl 135.11301]
http://www.emis.de/cgi-bin/Zarchive?an=0135.11301
[20] Adriaan C. Zaanen, AN INTRODUCTION TO THE THEORY OF
INTEGRATION, 2'nd edition, North-Holland, 1961.
[MR 20 #3950; Zbl 81.27703]
http://www.emis.de/cgi-bin/Zarchive?an=0081.27703
Dave L. Renfro
Paul Kornman
> Let {E_k}^{k=1}^\infty be a sequence of measurable subsets in [0,1]
> such that m(E_k) = 1 for all k. Show m(\bigcap_{k=1}^\infty E_k) = 1.
m(\bigcup_{k=1}^\infty E_k^c) \leq \sum_{k = 1}^{\infty}{m(E_k^c)} = 0,
so that
m(\bigcap_{k=1}^\infty E_k) = 1 - m(\bigcup_{k=1}^\infty E_k^c) = 1 - 0 = 1