HISTORICAL ESSAY ON F_SIGMA LEBESGUE NULL SETS
Dave L. Renfro
I'd post a write-up I did this morning of a talk I gave at a job
interview a couple of days ago. Most hard core analysts are
probably not going to find anything new in sections I and II, but
sections III and IV might be of interest.
####################################################################
####################################################################
I. STRENGTHENING THE RIEMANN INTEGRABILITY CONTINUITY CONDITION
We begin with a couple of definitions.
LEBESGUE NULL: Can be covered by countably many open intervals,
the sum of whose lengths can be arbitrarily small.
FIRST CATEGORY: Can be expressed as a countable union of nowhere
dense sets.
Each of these collections forms a sigma-ideal (closed under
subsets and countable unions). Since the intersection of any two
sigma-ideals is again a sigma-ideal, the collection of sets
simultaneously Lebesgue null and first category forms a
sigma-ideal. Moreover, because the real line can be written as
the union of a Lebesgue null set and a first category set, this
latter sigma-ideal is a strengthening of "small" that is
significantly stronger than what either of these two sigma-ideals
separately provides for.
R is the set of real numbers.
THEOREM 1-A: If f: R --> R is bounded and Riemann integrable, then
the set of points at which f is not continuous is
Lebesgue null.
PROOF: This result is well-known.
THEOREM 1-B: If f: R --> R is bounded and Riemann integrable, then
the set of points at which f is not continuous is
Lebesgue null and first category.
PROOF: The set of points of discontinuity of any function forms
an F_sigma set (can be written as a countable union of
closed sets), and it is easy to see that any F_sigma
Lebesgue null set is also first category. [Use the fact
that any closed Lebesgue null set has to be nowhere dense.]
JORDAN NULL: Can be covered by FINITELY many open intervals,
the sum of whose lengths can be arbitrarily small.
It can be shown that a set is Jordan null <==> its closure is
Lebesgue null. This is fairly well-known, although a bit difficult
to find in the literature. A proof can be found in [1].
[1] K. G. Johnson, "The sigma-ideal generated by the Jordan sets
in R^n", Real Analysis Exchange 19 (1993-94), 278-282.
[MR 95e:26004; Zbl 793.28002]
SIGMA-JORDAN NULL: Can be expressed as a countable union of Jordan
null sets.
Equivalently, a set is sigma-Jordan null <==> it can be covered by
countably many closed Lebesgue null sets <==> it can be covered by
an F_sigma Lebesgue null set.
Therefore, the collection of sigma-Jordan null sets is a sigma-ideal
that is contained in the sigma-ideal of sets simultaneously
Lebesgue null and first category.
####################################################################
####################################################################
II. A FIRST CATEGORY LEBESGUE NULL SET THAT ISN'T SIGMA-JORDAN NULL
THEOREM 2: There exists a set simultaneously Lebesgue null and
first category that is not sigma-Jordan null.
PROOF: Let C be a Cantor set having positive measure in each of
its portions. [That is, the intersection with C of every
R-neighborhood of each point of C has positive Lebesgue
measure. The standard constructions of positive measure
Cantor sets give rise to this "everywhere of locally
positive measure" property.]
Let H be a G_delta set in R such that:
(a) H is dense in C
(b) H is Lebesgue null
One way to obtain such a set is to let H be any Lebesgue
null set containing the endpoints of the complementary
intervals of C. [These endpoints are dense in C and form
a Lebesgue null set. By outer regularity of Lebesgue measure,
there exists a G_delta set in R containing these endpoints
that has the same Lebesgue measure as the set of these
endpoints.] Let
G = H intersect C.
I claim that G is a Lebesgue null and first category set
that is not sigma-Jordan null. Clearly, G is Lebesgue null
and first category. [G is nowhere dense in R, in fact.]
Now, for a contradiction, assume that
G is contained in UNION(n=1 to infinity) of F_n,
where each F_n is a closed Lebesgue null set.
First, note that the intersection of each F_n with C is
nowhere-dense-in-C. [If not, then F_n would be dense in
some portion of C. But F_n is closed (in R, and hence in C
as well), so F_n would contain a nonempty open-in-C set,
and hence F_n would have positive measure.] Therefore,
K = UNION(n=1 to infinity) of [ C intersect F_n ]
is first-category-in-C. However, C is Baire space (C is a
closed subset of R, hence C is complete under the same
metric, hence the Baire category theorem holds for C) and
K contains a dense G_delta-in-C set (namely, G), and so
K cannot be first-category-in-C, a contradiction.
REMARK 1: The same proof (use outer regularity of Hausdorff
h-measure) shows that there exists, for any Hausdorff
measure function h, a set G such that
(a) G has Hausdorff h-measure zero
and (b) G is not sigma-Jordan null.
Thus, no matter how small in the sense of Hausdorff
measure you may have proved some exceptional set is,
if you can show that your set is sigma-Jordan null
(equivalently, is contained in an F_sigma set having
the same Hausdorff h-measure), then you'll have a
strictly stronger result.
REMARK 2: By taking products with R^{n-1} and using the Baire
category theorem in same way, THEOREM 2 (indeed, even
the result in REMARK 1) holds in any Euclidean space R^n.
THEOREM 1-C: If f: R --> R is bounded and Riemann integrable, then
the set of points at which f is not continuous is
sigma-Jordan null. [This result is sharp.]
The proof I gave of THEOREM 2 can be found in several papers
published in recent years:
[2] Marek Balcerzak, James E. Baumgartner, and Jacek Hejduck,
"On certain sigma-ideals of sets", Real Analysis Exchange
14 (1988-89), 447-453. [MR 90m:28001; Zbl 679.28002]
[3] Robert D. Berman and Togo Nishiura, "Some mapping properties
of the radial-limit function of an inner function", J. London
Math. Soc. (2) 52 (1995), 375-390. [See corollary on p. 381.]
[MR 96m:30050; Zbl 835.30025]
[4] Zbigniew Grande, "Le rang de Baire de la famille de toutes les
fonctions ayant la propriete (K)", Fund. Math. 96 (1977), 9-15.
[See page 14.] [MR 57 #3327; Zbl 353.26003]
[5] Winfried Just and Claude Laflamme, "Classifying sets of measure
zero with respect to their open covers", Trans. Amer. Math. Soc.
321 (1990), 621-645. [See theorem 4.1 on page 627.]
[MR 91a:28003; Zbl 716.28003]
[6] Alexander S. Kechris and Alain Louveau, "Descriptive set theory
and harmonic analysis, J. Symbolic Logic 57 (1992), 413-441.
[See corollary 3 on page 425.] [MR 93e:04001; Zbl 766.03026]
[7] Bernd Kirchheim, "Solution of two problems concerning F-sigma
sets of measure zero", Real Analysis Exchange 16 (1990-91),
279-283. [See proposition 1.] [MR 92a:28003; Zbl 726.28002]
[8] Gyorgy Petruska, "On Borel sets with small cover: a problem of
M. Laczkovich", Real Analysis Exchange 18 (1992-93), 330-338.
[Errata: RAE 19, page 58.] [MR 95g:28003ab; Zbl 783.28001]
####################################################################
####################################################################
III. SOME HISTORICAL REMARKS ON SIGMA-JORDAN NULL SETS
The first to give such an example seems to be Frink in 1933 (see
pp. 524-525 of Frink [10]). His example was in R^2 and involved
the use of a measure preserving embedding of [0,1] into R^2
(i.e. a measure preserving Jordan curve). He remarked that it
appeared difficult to find such an example in R. Apparently Frink
believed that such an example existed but was unable to construct
one.
In 1958 Marcus (theorem 1 of Marcus [13]) gave a straightforward
proof, which he states that Paul Erdos is partly responsible for,
of such an example in R. After the proof he remarks (correctly)
that the same technique can be used to prove that such sets exist
in R^n for $n \geq 1$. The proof that Marcus gave is similar to
the proof I gave in THEOREM 2 above, although an error corrected
after publication (see MR 22 #12180) slipped in. For the set C,
Marcus used a first category set in [0,1] having a Lebesgue null
complement relative to [0,1]. However, it is easy to see that no
dense first category subset of [0,1] can be of second category in
itself. The proof he gave is corrected simply by requiring C
(A, in Marcus' paper) to be a Cantor set having positive measure
in each of its portions. This error was observed and corrected
by Trohimcuk [18] in 1961.
In addition, Trohimcuk observed that examples in R^n for $n \geq 2$
are immediate by forming the Cartesian product with R^{n-1}.
Interestingly, while Trohimcuk corrects the first part of the
proof given by Marcus [NOTE: Marcus was aware of Trohimcuk's paper,
as he was the author of its Zbl review.], nearly half of Trohimcuk's
paper is tied up with a lengthy proof for the existence of a
residual-in-C Lebesgue null set G, whose existence was correctly
obtained by Marcus. For such a set, Marcus simply cited the main
result in Marczewski/Sikorski [12] (see also theorem 16.5 on p. 64
of Oxtoby [17]), a paper perhaps unavailable to Trohimcuk.
In 1962 Marcus [14] generalized his earlier result to the setting
of an arbitrary Polish space equipped with a complete non-atomic
sigma-finite measure mu defined on the mu-completion of the Borel
sets. But to do this Marcus had to also assume the existence of a
perfect nowhere dense set each of whose nonempty open neighborhoods
has positive mu-measure. This latter assumption was shown by Darst
and Zink [9] in 1965 to be automatic from the other assumptions as
long as there exists a set having finite nonzero mu-measure.
Incidentally, the main point of Darst and Zink's paper was to
answer a question posed by Marcus concerning the limitations on
the Borel type that a first category Lebesgue null and not
sigma-Jordan null set can be (besides not being F_sigma). They
show that for each higher Borel class there exists an example
in that Borel class that doesn't belong to any lower Borel class.
Apparently independent of the above, Moszner [16] gave in 1966
essentially the same proof of a first category Lebesgue null, but
not sigma-Jordan null set that I gave above. He gives credit to
E. Marczewski for this construction, from whom he must have learned
it after his announcement in Moszner [15], since the possibility
of such a set was not raised in Moszner's earlier paper [15].
Finally, the construction of such a set is briefly outlined by
Lipinski [11] in 1972, the paper Balcerzak, Baumgartner, and Hejduk
[2] refer to for such a set. Although no references to earlier
constructions of such sets are given in Lipinski [11], it seems that
Lipinski has also known of their existence for some time. Lipinski
has informed me (in a handwritten letter dated Nov. 3, 1994)
that after he had proved a certain exceptional set involving the
differentiability of jump functions was an F_sigma Lebesgue null set
(published in 1957), Marczewski had asked him (in 1957) if any set
of measure zero is contained in such a set [NOTE: This question is
quite natural, since Marczewski had published in 1955 a proof that
any set of measure zero is contained in the set of points of
nondifferentiability of some monotone function, and Lipinski's 1957
paper studied the set of points where a monotone jump function has
an infinite derivative.] Lipinski answered orally with the
counterexample of a dense G_delta subset of a measure dense
Cantor set. Thus, it appears that the proof given by Moszner,
and attributed by him to Marczewski, may have originated from
Lipinski.
[9] Richard B. Darst and Robert E. Zink, "On a note of Marcus
concerning a problem posed by Frink", Proc. Amer. Math. Soc.
16 (1965), 926-928. [MR 31 #4872; Zbl 145 (pp. 53-54)]
[10] Orrin Frink, "Jordan measure and Riemann integration", Annals
Math. (2) 34 (1933), 518-526.
[Zbl 7 (p. 155); JFM 59 (pp. 260-261)]
[11] Jan S. Lipinski, "On derivatives of singular functions", Bull.
Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972),
625-628. [MR 48 #6335; Zbl 241.26007]
[12] Edward Marczewski [Szpilrajn] and Roman Sikorski, "Remarks on
measure and category", Colloq. Math. 2 (1949), 13-19.
[MR 12 (p. 398); Zbl 38 (p. 201)]
[13] Solomon Marcus, "Remarques sur les fonctions integrables au
sens de Riemann", Bull. Math. Soc. Sci. Math. Phys. R. P.
Roumaine (N.S.) 2 (50) (1958), 433-439.
[MR 22 #12180; Zbl 93 (p. 59)]
[14] Solomon Marcus, "On a problem posed by O. Frink Jr."
(Romanian), Com. Acad. R. P. Romine 12 (1962), 281-286.
[MR 26 #3849; Zbl 128 (p. 52)
[15] Zenon Moszner, "Remarques sur une notion de rarefaction d'un
ensemble de mesure nulle", C. R. Acad. Sci. Paris 256 (1963),
3556-3559. [MR 27 #255; Zbl 115 (p. 51)]
[16] Zenon Moszner, "Sur une notion de la rarefaction d'un ensemble
de mesure nulle", Ann. Sci. Ecole Norm. Sup. (3) 83 (1966),
191-200. [MR 36 #3658; Zbl 153 (p. 87)]
[17] John C. Oxtoby, MEASURE AND CATEGORY, 2'nd ed., Graduate Texts
in Math. 2, Springer-Verlag, 1980, x + 106 pages.
[MR 81j:28003; Zbl 435.28011]
[18] Ju. Ju. Trohimcuk, "An example of a point set" (Russian),
Ukrain. Mat. Zurn. 13 (1961), 117-118. [I have prepared a LaTeX
English translation of this paper.]
[MR 24 #A3252; Zbl 115 (p. 271)]
####################################################################
####################################################################
IV. SOME APPLICATIONS OF SIGMA-JORDAN NULL SETS
There are a some results involving sigma-Jordan null exceptional
sets for a variation of the two-dimensional Riemann integral in
Goldman [25]. [Rather than taking Riemann sums of rectangular
partitions whose maximal side lengths approach zero, Goldman uses
a modification proposed by Solomon Marcus in which the maximal
areas approach zero.]
Theorem 13.4 on page 51 of Oxtoby's book [17] states that a subset
E of R is first category <==> there exists a homeomorphism of R
onto itself such that the image of E is sigma-Jordan null.
In [24] Goffman proves that a subset E of R is sigma-Jordan null
<==> there exists a measurable set F such that the metric density
of F exists and is different from 0 or 1 at each point of E.
[NOTE: Goffman explicitly states the "only if" part as his
theorem 3, but in his theorem 2 he only states "first category
Lebesgue null" for the "if" part. However, an examination of the
proof he gives of theorem 2 shows that the set Z he proves to be
first category Lebesgue null is in fact contained in an F_sigma
Lebesgue null set.]
In [28] Mauldin proves that there exists a Borel map f from [0,1]
onto the Hilbert cube [0,1]^w such that for each x in [0,1]^w,
the inverse image of x under f is not sigma-Jordan null. (In fact,
f is of Borel class 3.)
Recall that every measurable set is the union of a Borel set with a
Lebesgue null set. Mauldin [29] shows that in R^n we cannot
strengthen "Lebesgue null" to "sigma-Jordan null", even if
"measurable set" is replaced with "analytic set".
Sigma-Jordan null sets play an important role in the study of
cardinal invariants. See Brendle [19], Bukovsky, Kholshchevnikova,
and Repicky [20], and Repicky [30] [31] [32].
Sigma-Jordan null sets also play an important role in the Borel
rarefaction method of classifying sets of measure zero in R.
See page 2034 of Frechet [21], page 236 of Frechet [22],
pages 161-165 of Frechet [23], Moszner [15], pages 192-195 of
Moszner [16], as well as the more recent papers [26] and [27]
by Laflamme.
Finally, the collection of sigma-Jordan null sets plays a crucial
role in Berman and Nishiura [3], in Grande [4], and in Pu/Pu [33].
[19] Jorg Brendle, "The additivity of porosity ideals", Proc. Amer.
Math. Soc. 124 (1996), 285-290. [MR 96d:04001; Zbl 839.03029]
[20] Lev Bukovsky, Natasha N. Kholshchevnikova, and Miroslav
Repicky, "Thin sets of harmonic analysis and infinite
combinatorics", Real Analysis Exchange 20 (1994-95), 454-509.
[MR 97b:43004; Zbl 835.42001]
[21] Maurice Frechet, "Sur la comparaison des rarefactions",
C. R. Acad. Sci. Paris 255 (1962), 2033-2036.
[MR 26 #1411; Zbl 109 (p. 279)]
[22] Maurice Frechet, "Sur la rarefaction d'un ensemble de mesure
nulle", Rend. Circ. Math. Palermo (2) 12 (1963), 229-238.
[MR 29 #204; Zbl 125 (p. 31)]
[23] Maurice Frechet, "Les probabilites nulles et la rarefaction",
Ann. Sci. Ecole Norm. Sup. (3) 80 (1963), 139-172.
[MR 28 #5450; Zbl 119 (p. 54)]
[24] Casper Goffman, "On Lebesgue's density theorem, Proc. Amer.
Math. Soc. 1 (1950), 384-388. [MR 12 (p. 167); Zbl 38 (p. 38)]
[25] Alan J. Goldman, "A variant of the two-dimensional Riemann
integral", J. Research National Bureau of Standards
Sect. B 69B (1965), 185-188. [MR 32 #4242; Zbl 136 (p. 349)]
[26] Claude Laflamme, "Some possible covers of measure zero sets",
Colloq. Math. 63 (1992), 211-218. [MR 93i:03072; Zbl 767.03025]
[27] Claude Laflamme, "A few sigma-ideals of measure zero sets
related to their covers", Real Analysis Exchange 17 (1991-92),
362-370. [MR 93b:28010; Zbl 764.03019]
[28] R. Daniel Mauldin, "The Baire order of the functions continuous
almost everywhere", Proc. Amer. Math. Soc. 41 (1973), 535-540.
[MR 48 #2319; Zbl 306.26004]
[29] R. Daniel Mauldin, Analytic non-Borel sets modulo null sets",
AMS Contemporary Math. 192 (1996), 69-70.
[MR 97c:28003; Zbl 840.28001]
[30] Miroslav Repicky, "Porous sets and additivity of Lebesgue
measure", Real Analysis Exchange 15 (1989-90), 282-298.
[MR 91a:03098; Zbl 716.28004]
[31] Miroslav Repicky, "Additivity of porous sets", Real Analysis
Exchange 16 (1990-91), 340-343. [92a:26011; Zbl 725.54007]
[32] Miroslav Repicky, Cardinal invariants related to porous sets",
pp. 433-438 in SET THEORY OF THE REALS ed. by H. Judah,
Israel Math. Conf. Proc. 6, 1993, viii + 654 pages.
[MR 94h:03095; Zbl 828.04001]
[33] Huo Hui Min Pu and Hwang-Wen Pu, "On the first class of Baire
generated by continuous functions on R^n relative to the
almost Euclidean topology", Acta Math. Hung. 57 (1991),
191-196. [MR 93e:26008; Zbl 747.26007]
####################################################################
####################################################################
Dave L. Renfro