I don't yet have a reference, but it seems that the result was 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 copy of Sikorski's book or simply a scan of the relevant page(s).