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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:

    (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).

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    $\begingroup$ Monotone real-valued functions on hyperspaces of continua are called Whitney maps (see any textbook on Continuum Theory). Maybe this will help? $\endgroup$ Sep 25, 2015 at 17:52
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    $\begingroup$ Possibly of interest: [A] Utpal Kumar Bandyopadhyay, On vector measures with the Darboux property, Quarterly Journal of Mathematics (Oxford) (2) 25 (1974), 57-60; [B] Hwang-Wen Pu and Huay-Min Huoh, Darboux property for transformations, Journal of Mathematical Analysis and Applications 90 #2 (December 1982), 299-306. $\endgroup$ Sep 25, 2015 at 18:25
  • $\begingroup$ @Dave L Renfro. Thanks. From what I can say, [B] is essentially focused on functions from $\mathbf R^n$ to (the ground set of) a metric space $\mathcal X$, for which the authors introduce, and study, a kind of Darboux property that is ultimately shaped by the Euclidean structure of the domain and the metric structure of $\mathcal X$. Not really what I'm looking for. (I don't have access to [A], right now.) $\endgroup$ Sep 25, 2015 at 19:03
  • $\begingroup$ I'm a little unclear what you're seeking with Q1. For finitely additive measures $f$, doesn't weak Darboux immediately imply strong Darboux, just by finding a subset of $Y \setminus X$ having measure $a-f(X)$? $\endgroup$ Sep 26, 2015 at 3:17
  • $\begingroup$ @Nate Eldredge. The question is a little bit unclear, indeed. I'm basically interested in understanding if (ii) has been called under a different name in the literature (either on measures or on densities). To put it in other terms, Q1 is more about keywords I can use to track the literature on the subject, whereas Q2 is more about actual results related to properties (i) and (ii). And again, I'm mostly interested in the case of densities, for which I couldn't really find that much. $\endgroup$ Sep 26, 2015 at 6:43

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