# A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows:

Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ and $a \in [0, f(X)]$ there exists $A \in \Sigma$ such that $A \subseteq X$ and $f(A) = a$.

In an attempt of finding out where this was first proved, I bumped into the Wiki article on atomic measures (here), which sketches a proof of a stronger result and provides a reference to [3].

However, the reference doesn't seem to be correct, as the paper above proves, to the best of my understanding, that:

Theorem 2. If $E_0$ is a bounded subset of ${\bf R}^m$ and $f$ a finitely additive function $\mathcal P(E_0) \to \bf R$ that is continue (**), then for all $E_1, E_2 \subseteq E_0$ and every $a \in [0,1]$ there exists a (Lebesgue-)measurable set $A \subseteq E_0$ such that $f(A) = a f(E_1) + (1-a) f(E_2)$.

(Caveat lector: I'm essentially sticking to Sierpiński's notation and terminology, and hoping to interpret them correctly.) So my question is:

Q. Assuming that I'm not missing something, do you know of a correct reference to Sierpiński's work where Theorem 1 was first proved?

Update (Nov 04, 2015). It follows from [2, Theorem 1] that the range of a countably additive non-atomic vector measure $\mu:\Sigma\to{\bf R}^n$ (viz., an $n$-tuple of countably additive measures $\Sigma\to\bf R$ each of which is non-atomic) is convex. This, yet, doesn't seem to have a clear relation with Theorem 1 above, as it implies that for all $X\in\Sigma$ and $a\in [0,1]$ there exists $A\in\Sigma$ s.t. $\mu(A)=a\mu(X)$, while for Theorem 1 we should also have $A\subseteq X$. Lyapunoff's theorem appears as (part of) Theorem 13 in [1, Chapter 2].

In particular, it is noted on p. 39 of Fryszkowski's book that "There is a long story of results concerning the range of vector measures. The first result of this type belongs to Sierpiński [...]," and here a reference is given to the 1922 paper I cited before, "who showed that the range of a real non-atomic measure is a compact interval." This, however, is not the content of Theorem 2 above, is it?! So I feel all the more confused by the entire story, and continue missing the link with Theorem 1 above.

Notes.

(*) It means that if $X \in \Sigma$ and $|f(X)| > 0$ then there exists $Y \in \Sigma$ such that $Y \subseteq X$ and $0 < |f(Y)| < |f(X)|$.

(**) It means that $f(E) \to 0$ as $E \subseteq E_0$ and the diameter of $E$ tends to $0$.

Bibliography.

[1] A. Fryszkowski, Fixed Point Theory for Decomposable Sets, Topological Fixed Point Theory and Its Applications 2, Dordrecht: Kluwer Academic Publishers, 2004.

[2] A. A. Liapounoff, Sur les fonctions-vecteurs complètement additives, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), No. 6, 465-478.

[3] W. Sierpiński, Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), No. 1, 240–246 (in French).

• For the record: Theorem 1 is mentioned as well in a 2014 question by @MannyReyes: mathoverflow.net/questions/187975, which also makes reference to the same Wikipedia article as in the OP. – Salvo Tringali Nov 3 '15 at 22:22
• Incidentally, the proof of the theorem (and, more specifically, its dependency on the axiom of choice) is discussed here: mathoverflow.net/questions/225677/…. – Salvo Tringali Dec 10 '15 at 9:01
• Doesn't one get thm 1 from thm 2 simply by picking some $E_2$ that has measure zero? It says "for all" and so surely I can pick something? Is this question secretly about the axiom of choice, or am I missing something that was supposed to be obvious? – Linas Sep 13 '20 at 5:58

I don't yet have a reference, but it seems the result might have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:

R. Sikorski, Real Functions, Vol. 1, PWN: Warsaw, 1958 (in Polish),

at least according to the historical remark on p. 28 in:

P. Lorenc and R. Wituła, Darboux property of the nonatomic $\sigma$-additive positive and finite dimensional vector measures, Matematyka Stosowana 3 (2013), 25-36.

Unfortunately, I couldn't retrieve either a hard copy of Sikorski's book or simply a scan of the relevant page(s).

Update (Nov 27, 2015). Thanks to Martin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.

Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz [${}$1]". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?

Update (Dec 08, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.

With the invaluable help of Nathalie Granottier and Fabienne Grosjean (Bibliothèque du CIRM, Institut de Mathématiques de Luminy) and Cyril Mauvillain (Bibliothèque de Recherche Mathématiques et Informatique, Université Bordeaux 1), I've finally got a scan copy of the bibliography of Sikorski's book.

The final outcome is that "Sierpiński [7]" (from the footnote on p. 225 of Sikorski's book) is nothing but the paper of Sierpiński that was already cited in the OP, namely

W. Sierpiński, Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), No. 1, 240–246 (in French),

while "Fichtenholz [${}$1]" is

G. Fichtenholz, Sur les fonctions d'ensemble additives et continues, Fund. Math. 7 (1925), No. 1, 296–301 (in French).

Moreover, neither of these two papers deals with Theorem 1 in the OP, and this is not in contradiction to Sirkoski's statement (though it is in contradiction to statements made in a bunch of places, where the result is attributed to Sierpiński and a reference is made to the above paper of his), as the text in the footnote on p. 225 of Sikorski's book says "por. także", which sounds more like a "see also", which I interpret as a "cf." (and makes perfect sense), than like a "see", which I would rather interpret as "the result appears in".

So, my conclusion is that, if anyone, then it's Sikorski who should deserve credit for the result, which in fact applies to countably additive measures (as of the statement in the OP), whereas measures considered in Sierpiński's and Fichtenholz' papers are finitely additive.

And to confirm that Sikorski is really talking of countably additive measures, let me cite from the incipit of Section 3, Chapter VI of his book:

§3. Miara. Zgodnie z uwagami wstępnymi z §1 miarą nazywamy każdą nieujemną, przeliczalnie addytywną funkcję zbioru $\mu$, określoną na pewnym przeliczalnie addytywnym ciele $\mathfrak M$ podzbiorów przestrzeni $X$, tzn. nieujemna funkcję rzeczywistą na $\mathfrak M$, nierówną tożsamościowy $\infty$, taką, że dla każdego ciągu nieskończonego zbiorów rozłącznych $A_n \in \mathfrak M$ $$\mu(A_1 + A_2 + \ldots) = \mu(A_1) + \mu(A_2) + \ldots$$

More or less, this should translate into:

§3. Measures. Based on §1, we let a measure be any nonnegative, countably additive set function $\mu$, defined on a sigma-algebra $\mathfrak M$ of subsets of $X$, i.e. a non-negative real function on $\mathfrak M$, possibly taking also the value $\infty$, such that for every countable sequence of disjoint sets $A_n \in \mathfrak M$ we have $$\mu (A_1 + A_2 + \ldots) = \mu (A_1) + \mu (A_2) + \ldots$$

• Unless something new pops up, I'm accepting the above as the answer to the question in the OP. – Salvo Tringali Dec 7 '15 at 22:39

P. R. Halmos, Measure Theory has in the back of the book references for its results. For exercise (8) in section 40, perhaps more general than the question here, he cites:

D. Maharam, "On homegeneous measure algebras," Proc. Nat. Acad. Sci. 29 (1942) 108-111.

Perhaps references in the Maharam paper will show where your exact result comes from.

• AFAICS, Maharam's paper deals with nontrivial Boolean lattices $\mathbb L = (L, \vee, \land)$ that are closed under countable joins (she calls them "Boolean $\sigma$-algebras") and endowed with a countably additive nonnegative measure $\mu:L\to\bf R$ s.t. $\mu(a)=0$ iff $a=0_{\mathbb L}$, where $0_{\mathbb L}$ is the least element of $\mathbb L$; she refers to $(\mathbb L,\mu)$ as a measure algebra. The paper has 3 theorems and 2 lemmas, which are mostly specialized to homogeneous measure algebras, as defined on the first lines of Section 2. But I don't see any connection with the OP. – Salvo Tringali Nov 4 '15 at 7:14
• I've just given a look at Exercise (8) in Sect. 8 of the 1974 Springer ed. of Halmos' Measure Theory. I can't really say at present whether or not it has something to do with the OP, but the ref. provided on p. 291 points to: E. Marczewski, Sur l'isomorphie des mesures séparables, Colloq. Math. 1 (1947), 39-40. Yet, this doesn't match with the db of Colloq. Math., where I could rather find the following: E. Marczewski, Indépendance d'ensembles et prolongement de mesures (Résultats et problèmes), Colloq. Math. 1 (1947), 122-132, which discusses Halmos' exercise in Sect. 1. – Salvo Tringali Nov 4 '15 at 8:06
• I haven't looked at Halmos' book but my guess is the following: Maharam's theorem is a characterization of measure algebras - Every atomless measure algebra is essentially a countable join of measure algebras of the product measures on $[0, 1]^{\kappa}$ from which your Theorem 1 immediately follows. – Ashutosh Nov 4 '15 at 22:57
• @Ashutosh. Gerard has probably misread the reference provided by Halmos in relation to Exercise (8) in Sect. 40 of his book (see my 2nd comment above, but change "Sect. 8" to "Sect. 40"). However, I agree with you on the rest: Maharam's characterization (Theorem 1 in her paper) applies only to homogeneous measure algebras, yet Theorem 2 in the same paper yields a decomposition of an arbitrary measure algebra as a (finite or countably infinite) direct sum of homogeneous measure algebras, and this can be used to prove Theorem 1 in the OP. Nonetheless, I'm not sure this answers my question. – Salvo Tringali Nov 7 '15 at 19:09