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Moreover, $\mathcal M(f^\ast)$ is always infinite, as any function of the form $\mathbf H \to \mathbf H: x \mapsto x+h$, with $h \in \bf N$, belongs to $\mathcal M(f^\ast)$, and the same is true of $\mathcal G(f^\ast)$ when $\mathbf H = \mathbf Z$ (regardless of whether or not $f^\ast$ is monotone), which marks a difference, as minor as it may be, among the cases parametrized by $\bf H$. On the other hand, it's completely unclear to me if there is any (substantial) difference between the cases ${\bf H} = \bf N$ and ${\bf H} = \bf N^+$, the question boiling down most probably to the existence of a unique extension of an upper quasi-density on ${\bf N}^+$ to a quasi-density on $\bf N$ (which is certainly true in the presence of monotonicity).

Moreover, $\mathcal M(f^\ast)$ is always infinite, as any function of the form $\mathbf H \to \mathbf H: x \mapsto x+h$, with $h \in \bf N$, belongs to $\mathcal M(f^\ast)$, and the same is true of $\mathcal G(f^\ast)$ when $\mathbf H = \mathbf Z$ (regardless of whether or not $f^\ast$ is monotone), which marks a difference, as minor as it may be, among the cases parametrized by $\bf H$. On the other hand, it's completely unclear to me if there is any (substantial) difference between the cases ${\bf H} = \bf N$ and ${\bf H} = \bf N^+$, when my feeling is that the question should boil down to deciding whether or not there exists a unique extension of an upper quasi-density on ${\bf N}^+$ to a quasi-density on $\bf N$ (which is certainly true in the presence of monotonicity).

Q1. Is $\mathcal M(f)$ infinite? Q2. Is $\mathcal G(f)$ infinite? Q3. Is $\mathcal G(f)$ a group under composition? Q4. And what about Q1 and Q2 with $\mathcal G(f^\ast)$ in place of $\mathcal G(f)$? (Of course, $\mathcal G(f^\ast)$ does always form a group under the operation of composition.)

Moreover, $\mathcal M(f^\ast)$ is always infinite, as any function of the form $\mathbf H \to \mathbf H: x \mapsto x+h$, with $h \in \bf N$, belongs to $\mathcal M(f^\ast)$, and the same is true of $\mathcal G(f^\ast)$ when $\mathbf H = \mathbf Z$ (which marks a difference, as minor as it may be, among the cases parametrized by $\bf H$), regardless of whether or not $f^\ast$ is monotone.

Q1. Is $\mathcal M(f)$ infinite? Q2. Is $\mathcal G(f)$ infinite? Q3. Is $\mathcal G(f)$ a group under composition? Q4. And what about Q1 and Q2 with $f^\ast$ in place of $f$? (Of course, $\mathcal G(f^\ast)$ does always form a group under the operation of composition.)

Moreover, $\mathcal M(f^\ast)$ is always infinite, as any function of the form $\mathbf H \to \mathbf H: x \mapsto x+h$, with $h \in \bf N$, belongs to $\mathcal M(f^\ast)$, and the same is true of $\mathcal G(f^\ast)$ when $\mathbf H = \mathbf Z$ (regardless of whether or not $f^\ast$ is monotone), which marks a difference, as minor as it may be, among the cases parametrized by $\bf H$. On the other hand, it's completely unclear to me if there is any (substantial) difference between the cases ${\bf H} = \bf N$ and ${\bf H} = \bf N^+$, the question boiling down most probably to the existence of a unique extension of an upper quasi-density on ${\bf N}^+$ to a quasi-density on $\bf N$ (which is certainly true in the presence of monotonicity).

We call $\mathcal M(f)$ the Lévy monoid of $f$, by analogy with what is usually called the Lévy group after Hida [H], who was in turn motivated by earlier work of Lévy [L, Part III] (I don't have access to a copy of Lévy's book, so this is second-hand information). In addition, we set $$\mathcal G(f) := \mathcal M(f) \cap {\rm aut}({\bf H}),$$ where ${\rm aut}({\bf H})$ is the set of all permutations of ${\bf H}$, and we refer to $\mathcal G(f)$ as the Lévy group of $f$ if, to nobody's surprise, $\mathcal G(f)$ is a subgroup of $\mathcal M(f)$.

Q1. Is $\mathcal M(f)$ infinite? Q2. Is $\mathcal G(f)$ infinite? Q3. Is $\mathcal G(f)$ a subgroup of $\sf Set$? Q4. And what about Q1 and Q2 with $\mathcal G(f^\ast)$ in place of $\mathcal G(f)$? (Of course, $\mathcal G(f^\ast)$ is always a subgroup of $\sf Set$.)

Both Q1 and Q2 can be answered in the affirmative if $f^\ast$ is monotone (nondecreasing), viz. $f^\ast(X) \le f^\ast(Y)$ whenever $X \subseteq Y \subseteq \bf H$ (in which case $f^\ast$ is called an upper density on $\bf H$ and $f$ an induced density).

Q4. What is there in the literature about $\mathcal G(f)$ in the special case where $f^\ast$ is, say, the upper Banach density, the upper logarithmic density, or the upper analytic density?

In spite of a few articles dealing with the Lévy group of the asymptotic density on $\bf N$ and/or measure densities extending the asymptotic density to a finitely additive probability measure $\mathcal P(\mathbf N) \to \bf R$, see e.g. [NP], [SZ] and references therein, I can't even mention a single paper picking up with any of the cases mentioned in Q4, can you?

We call $\mathcal M(f)$ the Lévy monoid of $f$, by analogy with what is usually called the Lévy group after Hida [H], who was in turn motivated by earlier work of Lévy [L, Part III] (I don't have access to a copy of Lévy's book, so this is second-hand information). In addition, we set $$\mathcal G(f) := \mathcal M(f) \cap {\rm aut}({\bf H}),$$ where ${\rm aut}({\bf H})$ is the set of all permutations of ${\bf H}$, and we refer to $\mathcal G(f)$ as the Lévy group of $f$ if, to nobody's surprise, $\mathcal G(f)$ is a subgroup of $\mathcal M(f)$, i.e. $f^{-1} \in \mathcal G(f)$ for every $f \in \mathcal G(f)$.

Q1. Is $\mathcal M(f)$ infinite? Q2. Is $\mathcal G(f)$ infinite? Q3. Is $\mathcal G(f)$ a group under composition? Q4. And what about Q1 and Q2 with $\mathcal G(f^\ast)$ in place of $\mathcal G(f)$? (Of course, $\mathcal G(f^\ast)$ does always form a group under the operation of composition.)

Both Q1 and Q2 can be answered in the affirmative if $f^\ast$ is monotone (nondecreasing), viz. $f^\ast(X) \le f^\ast(Y)$ whenever $X \subseteq Y \subseteq \bf H$, in which case $f^\ast$ is called an upper density on $\bf H$ and $f$ an induced density.

Q5. What is there in the literature about $\mathcal G(f)$ in the special case where $f^\ast$ is, say, the upper Banach density, the upper logarithmic density, or the upper analytic density (on $\bf N$)? (All of these are upper densities.)

For the record: In spite of a few articles dealing with the Lévy group of the asymptotic density on $\bf N$ and/or measure densities extending the asymptotic density to a finitely additive probability measure $\mathcal P(\mathbf N) \to \bf R$, see e.g. [NP], [SZ] and references therein, I can't even mention a single paper picking up with any of the cases mentioned in Q5.