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

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

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), 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 (in the sense that $f(E) \to 0$ as $E \subseteq E_0$ and the diameter of $E$ tends to $0$), 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 translating from the French, 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?