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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:

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

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 $a \in [f(E_1), f(E_2)] \cup [f(E_2), f(E_1)]$ there exists a (Lebesgue-)measurable set $A \subseteq E_0$ such that $f(A) = a$.

Update (Nov 04, 2015). It follows from Theorem 1 in:

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

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 $A\subseteq X$.

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 $0$.

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].

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 $f(A) = a f(E_1) + (1-a) f(E_2)$.

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).

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:

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

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 $a \in [f(E_1), f(E_2)] \cup [f(E_2), f(E_1)]$ there exists a (Lebesgue-)measurable set $A \subseteq E_0$ such that $f(A) = a$.

(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?

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 $0$.

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:

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

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 $a \in [f(E_1), f(E_2)] \cup [f(E_2), f(E_1)]$ there exists a (Lebesgue-)measurable set $A \subseteq E_0$ such that $f(A) = a$.

(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 Theorem 1 in:

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

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 $A\subseteq X$.

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 $0$.