There is a 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][1]), 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 sticking to Sierpiński's notation and terminology, and trying my best to interpret them in a correct way.) So my question is:

> **Q.** Assuming that I'm not missing something, do you know of a correct reference to Theorem 1?

**Notes.** (*) It means that if $X \subseteq S$ and $f(X) > 0$ then there exists $Y \subseteq X$ such that $0 < f(Y) < f(X)$. (**) It means that $f(E) \to 0$ as $E \subseteq E_0$ and the diameter of $E$ tends to $0$.

  [1]: https://en.wikipedia.org/wiki/Atom_%28measure_theory%29#Non-atomic_measures