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Update (Nov 27, 2015). Thanks to Martin SleziakMartin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.

Update (Nov 27, 2015). Thanks to Martin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.

Update (Nov 27, 2015). Thanks to Martin Sleziak and Jacek Jendrej, I found out that the "remark to Problem 12" referred to by Lorenc and Wituła is actually a footnote on p. 225 of Sikorski's book.

Fixed a couple of typos
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Salvo Tringali
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I don't yet have a reference, but it seems that the result wasmight have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:

Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz 1"[${}$1]". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?

Update (Dec 0708, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.

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:

Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz 1". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?

Update (Dec 07, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.

I don't yet have a reference, but it seems the result might have been first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:

Now, Problem 12 reads, "If $\mu$ is a non-atomic measure and $0 < s < \mu(A) < \infty$, then there exists $B \subseteq A$ such that $\mu(B) = s$", so we are really talking of Theorem 1 in the OP. And the footnote on p. 225 really makes reference to "Sierpiński [7] and Fichtenholz [${}$1]". But the copy of Sikorski's book I could retrieve is incomplete and doesn't include the bibliography... Could anyone having access to a complete copy fill in this answer?

Update (Dec 08, 2015). Here is some more evidence that the common (?) attribution of Theorem 1 in the OP to W. Sierpiński may be wrong.

Improved exposition and fixed a couple of typos
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Salvo Tringali
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This has very much confused me in the first placeAnd to confirm that Sikorski is really talking of countably additive measures, butlet me cite from the incipit of Section 3, Chapter VI of Sikorski'shis book reads:

And this should moreMore or less mean, this should translate into:

§3. Measures. Based on §1, we let a measure be any nonnegative, countably additive set function $\mu$, defined on a sigma-algebra $ \mathfrak M $ of subsets of $ X $, i.e. a non-negative real function on $\mathfrak M $, possibly taking also the value $ \infty $, such that for every countable sequence of disjoint sets $ A_n \in \mathfrak M $ we have $$ \mu (A_1 + a_2 + \ldots) = \mu (A_1) + \mu (a_2) + \ldots $$$$ \mu (A_1 + A_2 + \ldots) = \mu (A_1) + \mu (A_2) + \ldots $$

To wit, Sikorski is really talking of countably additive measures.

This has very much confused me in the first place, but the incipit of Section 3, Chapter VI of Sikorski's book reads:

And this should more or less mean:

§3. Measures. Based on §1, we let a measure be any nonnegative, countably additive set function $\mu$, defined on a sigma-algebra $ \mathfrak M $ of subsets of $ X $, i.e. a non-negative real function on $\mathfrak M $, possibly taking also the value $ \infty $, such that for every countable sequence of disjoint sets $ A_n \in \mathfrak M $ we have $$ \mu (A_1 + a_2 + \ldots) = \mu (A_1) + \mu (a_2) + \ldots $$

To wit, Sikorski is really talking of countably additive measures.

And to confirm that Sikorski is really talking of countably additive measures, let me cite from the incipit of Section 3, Chapter VI of his book:

More or less, this should translate into:

§3. Measures. Based on §1, we let a measure be any nonnegative, countably additive set function $\mu$, defined on a sigma-algebra $ \mathfrak M $ of subsets of $ X $, i.e. a non-negative real function on $\mathfrak M $, possibly taking also the value $ \infty $, such that for every countable sequence of disjoint sets $ A_n \in \mathfrak M $ we have $$ \mu (A_1 + A_2 + \ldots) = \mu (A_1) + \mu (A_2) + \ldots $$

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Salvo Tringali
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Updated and probably solved
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Updated and probably solved
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Update
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