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 $f$ afinitelyadditive function $\mathcal P(E_0) \to \bf R$ that iscontinue(**), then for all $E_1, E_2 \subseteq E_0$ and every $a \in [0,1]$ there exists a(Lebesgue-)measurableset $A \subseteq E_0$ 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).