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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 $D$ the domain of $f$. We say that $f$ has:

(i) the weak Darboux property if $\emptyset \in D$ and for every $X \in D$ and $a \in [f(\emptyset),f(X)]$ there exists $A \in D$ such that $A \subseteq X$ and $f(A) = a$;



(ii) the (strong) Darboux property if for all $X, Y \in D$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists $A \in D$ such that $X \subseteq A \subseteq Y$ and $f(A) = a$.

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 $X \subseteq S$.

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 $f(A) = a$.

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 $X \in \mathcal D$.

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$. We say that $f$ has:

(i) the weak Darboux property if for every $X \subseteq S$ and $a \in [f(\emptyset),f(X)]$ there exists $A \in {\rm dom}(f)$ contained in $X$ such that $f(A) = a$;



(ii) the (strong) Darboux property if for all $X, Y \subseteq S$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists a set $A \in {\rm dom}(f)$ containing $Y$ and contained in $X$ such that $f(A) = a$.

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 in 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] and [5] points to [4] as a source for the terminology, and [3] points to [5]), 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.

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" (as interpreted, say, in additive and probabilistic number theory). Do you have any pointer to suggest?

And my second question (which partially overlaps with Q1) is:

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 $D$ the domain of $f$. We say that $f$ has:

(i) the weak Darboux property if $\emptyset \in D$ and for every $X \in D$ and $a \in [f(\emptyset),f(X)]$ there exists $A \in D$ such that $A \subseteq X$ and $f(A) = a$;



(ii) the (strong) Darboux property if for all $X, Y \in D$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists $A \in D$ such that $X \subseteq A \subseteq Y$ and $f(A) = a$.

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.

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:

• (i) the weak Darboux property if for every $X \subseteq S$ and $a \in [f(\emptyset),f(X)]$ there exists $A \in {\rm dom}(f)$ contained in $X$ such that $f(A) = a$;
• (ii) the (strong) Darboux property if for all $X, Y \subseteq S$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists a set $A \in {\rm dom}(f)$ containing $Y$ and contained in $X$ such that $f(A) = a$.

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 $X \subseteq S$, 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.

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 in 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] and [5] points to [4] as a source for the terminology, whereas [3] points to [5]), 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.

And my second question is:

(i) the weak Darboux property if for every $X \subseteq S$ and $a \in [f(\emptyset),f(X)]$ there exists $A \in {\rm dom}(f)$ contained in $X$ such that $f(A) = a$;



(ii) the (strong) Darboux property if for all $X, Y \subseteq S$ with $X \subseteq Y$ and every $a \in [f(X),f(Y)]$ there exists a set $A \in {\rm dom}(f)$ containing $Y$ and contained in $X$ such that $f(A) = a$.

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 in 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] and [5] points to [4] as a source for the terminology, and [3] points to [5]), 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.

And my second question (which partially overlaps with Q1) is: