Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
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(i) the weak Darboux property if $\emptyset \in \mathcal D$ and for every $X \in \mathcal D$ and $a
\in [f(\emptyset),f(X)]$ there exists $A \in \mathcal D$ such that $A \subseteq X$ and $f(A) = a$;
(ii) the (strong) Darboux property if for all $X, Y \in \mathcal D$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists $A \in \mathcal D$ such that $X \subseteq A \subseteq Y$ and $f(A) = a$.
It is perhaps worth remarking that, since we do not assume $f$ to be monotone in the above formulation of the weak Darboux property, it may well happen that $f(X) < f(\emptyset)$ for some $X \in \mathcal D$, in which case the interval $[f(\emptyset), f(X)]$ is empty and the property is vacuously true; analogous considerations apply to the Darboux property, too.
Of course, $f$ has the Darboux property only if it has the weak Darboux property, and has in turn the weak Darboux property only if $f(X) \le f(\emptyset)$ for every finite $X \in \mathcal D$.
Some authors, either in the area of measure theory, see, e.g., [4, Chapter V, Section 46.I, Corollary 3${}^\prime$] and [2, Chapter I, Section 2.9, Definition 4], or in connection to the study of "densities", say, in additive and probablistic number theory, see, e.g., [6, Section 2], [5, p. 217], and [3], refer to (i), and not to (ii), as the Darboux property (note that [6] points to [2], [5] points to [4], and [3] points to [5] as a source for the terminology), but that does not sound very fit to me, as (ii) is arguably closer than (i) to the spirit of the intermediate value property of real-valued functions of one real variable, so let me stick to my own definitions above.
Another term of common usage to allude to condition (i), particularly in the literature on charges, is ``strongly non-atomic'', see, e.g., [1, Definition 5.1.5].
Q1. I could not find any occurence of (ii) either in the literature on (finitely or countably additive, signed or unsigned, bounded or unbounded) measures, or in the literature on densities. Do you have any pointer to suggest?
In principle, I'm more interested in densities than in measures, but still... And my second question (which partially overlaps with Q1) is:
Q2. Do you know of other (nontrivial) results about measures (including charges) and densities that are related to the weak or strong Darboux properties (apart from those referred to in this post)?
As for Q2, I would certainly include a theorem of W. Sierpiński [7], which proves the weak Darboux property of non-atomic (countably additive) probability measures.
Bibliography.
[1] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges: A Study of Finitely Additive Measures, Pure and Applied Mathematics 109, London: Academic Press, 1983.
[2] N. Dinculeanu, Vector Measures, International Series of Monographs in Pure and Applied Mathematics 95, Oxford: Pergamon Press, 1966.
[3] G. Grekos, L. Mišík, and J. T. Tóth, Density sets of sets of positive integers, J. Number Theory 130 (2010), No. 6, 1399-1407.
[4] K. Kuratowski, Topology: Volume I, London: Academic Press, 1966.
[5] M. Mačaj, L. Mišík, and J. Tomanová, On a class of densities of sets of positive integers, Acta Math. Univ. Comenianae 72 (2003), No. 2, 213-221.
[6] M. Paštéka and T. Šalát, Buck's measure density and sets of positive integers containing arithmetic progression, Math. Slovaca 41 (1991), No. 3, 283-293.
[7] W. Sierpinski, Sur les fonctions d'ensemble additives et continues, Fund. Math. 3 (1922), 240-246 (in French).
