Update: I am in the bad situation that I cannot make a paper because of this question. A positive answer would imply a complete solution of a problem in game theory, making a paper; a negative solution would imply that the results that I already have are the best possible and so it would make a paper. Taking inspiration from Erdos and also from an old MO question (that I can't find again), I put a real bounty of 100 euros over this question. The answer might be even mathematically interesting in itself, making a paper by itself - that would be great! Also private communication are possible (you can find my email in my profile).

Let $\mathcal M(\mathbb Z)$ be the set of all finitely additive probability measures on the power set of $\mathbb Z$. Let $\phi:\mathbb Z\rightarrow\mathbb R$ be nonnegative and bounded. Observe that $\phi$ is integrable with respect to any $\mu\in\mathcal M(\mathbb Z)$. Let me say that $\mu$ is $\phi$-translation invariant if for all $y\in\mathbb Z$ one has

$$ \int\phi(x+y)d\mu(x)=\int\phi(x)d\mu(x) $$

Let $I_\phi(\mathbb Z)$ be the class of $\phi$-invariant measures in $\mathcal M(\mathbb Z)$. Let $\mu\in I_\phi(\mathbb Z)$ be fixed.

Question: Is it true that the mapping $F:\nu\in\mathcal M(\mathbb Z)\rightarrow\int\int\phi(x+y)d\nu(x)d\mu(y)$ attains its maximum on a measure $\nu\in I_\phi(\mathbb Z)$?

Update: It was observed by Pietro Majer that it is not clear that such a maximum should exist, since $F$ could not be continuous in the weak* topology. Even more, it seems to me that continuity of $F$ w.r.t. the weak* topology, implies that one can change the order of integration, which is not the case, at least in general. But, let me discuss briefly an example, to show why I believe that such maximum should exist and attained on an invariant measure: let $\phi$ be the characteristic function of $\mathbb N\subseteq\mathbb Z$. In this case $F$ is not continuous, but it is clear that the maximum is attained in a invariant measure on $\mathbb Z$ which is concentrated in $\mathbb N$.

Thanks in advance,

Valerio

Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.

Let $\mathcal M(\mathbb Z)$ be the set of all finitely additive probability measures on the power set of $\mathbb Z$. Let $\phi:\mathbb Z\rightarrow\mathbb R$ be nonnegative and bounded. Observe that $\phi$ is integrable with respect to any $\mu\in\mathcal M(\mathbb Z)$. Let me say that $\mu$ is $\phi$-translation invariant if for all $y\in\mathbb Z$ one has

$$ \int\phi(x+y)d\mu(x)=\int\phi(x)d\mu(x) $$

Let $I_\phi(\mathbb Z)$ be the class of $\phi$-invariant measures in $\mathcal M(\mathbb Z)$. Let $\mu\in I_\phi(\mathbb Z)$ be fixed.

Question: Is it true that the mapping $F:\nu\in\mathcal M(\mathbb Z)\rightarrow\int\int\phi(x+y)d\nu(x)d\mu(y)$ attains its maximum on a measure $\nu\in I_\phi(\mathbb Z)$?

Thanks in advance,

Valerio

This might be a stupid question. It can be clearly asked in much more generality (amenable semigroups), but I guess everything will be clear already in the case of the integers.

Let $\mathcal M(\mathbb Z)$ be the set of all finitely additive probability measures on the power set of $\mathbb Z$. Let $\phi:\mathbb Z\rightarrow\mathbb R$ be nonnegative and bounded. Observe that $\phi$ is integrable with respect to any $\mu\in\mathcal M(\mathbb Z)$. Let me say that $\mu$ is $\phi$-translation invariant if for all $y\in\mathbb Z$ one has

$$ \int\phi(x+y)d\mu(x)=\int\phi(x)d\mu(x) $$

Let $I_\phi(\mathbb Z)$ be the class of $\phi$-invariant measures in $\mathcal M(\mathbb Z)$. Let $\mu\in I_\phi(\mathbb Z)$ be fixed.

Question: Is it true that the mapping $F:\nu\in\mathcal M(\mathbb Z)\rightarrow\int\int\phi(x+y)d\nu(x)d\mu(y)$ attains its maximum on a measure $\nu\in I_\phi(\mathbb Z)$?

Update: It was observed by Pietro Majer that it is not clear that such a maximum should exist, since $F$ could not be continuous in the weak* topology. Even more, it seems to me that continuity of $F$ w.r.t. the weak* topology, implies that one can change the order of integration, which is not the case, at least in general. But, let me discuss briefly an example, to show why I believe that such maximum should exist and attained on an invariant measure: let $\phi$ be the characteristic function of $\mathbb N\subseteq\mathbb Z$. In this case $F$ is not continuous, but it is clear that the maximum is attained in a invariant measure on $\mathbb Z$ which is concentrated in $\mathbb N$.

Thanks in advance,

Valerio

Update: I am in the bad situation that I cannot make a paper because of this question. A positive answer would imply a complete solution of a problem in game theory, making a paper; a negative solution would imply that the results that I already have are the best possible and so it would make a paper. Taking inspiration from Erdos and also from an old MO question (that I can't find again), I put a real bounty of 100 euros over this question. The answer might be even mathematically interesting in itself, making a paper by itself - that would be great! Also private communication are possible (you can find my email in my profile).

Let $\mathcal M(\mathbb Z)$ be the set of all finitely additive probability measures on the power set of $\mathbb Z$. Let $\phi:\mathbb Z\rightarrow\mathbb R$ be nonnegative and bounded. Observe that $\phi$ is integrable with respect to any $\mu\in\mathcal M(\mathbb Z)$. Let me say that $\mu$ is $\phi$-translation invariant if for all $y\in\mathbb Z$ one has

$$ \int\phi(x+y)d\mu(x)=\int\phi(x)d\mu(x) $$

Let $I_\phi(\mathbb Z)$ be the class of $\phi$-invariant measures in $\mathcal M(\mathbb Z)$. Let $\mu\in I_\phi(\mathbb Z)$ be fixed.

Question: Is it true that the mapping $F:\nu\in\mathcal M(\mathbb Z)\rightarrow\int\int\phi(x+y)d\nu(x)d\mu(y)$ attains its maximum on a measure $\nu\in I_\phi(\mathbb Z)$?

Update: It was observed by Pietro Majer that it is not clear that such a maximum should exist, since $F$ could not be continuous in the weak* topology. Even more, it seems to me that continuity of $F$ w.r.t. the weak* topology, implies that one can change the order of integration, which is not the case, at least in general. But, let me discuss briefly an example, to show why I believe that such maximum should exist and attained on an invariant measure: let $\phi$ be the characteristic function of $\mathbb N\subseteq\mathbb Z$. In this case $F$ is not continuous, but it is clear that the maximum is attained in a invariant measure on $\mathbb Z$ which is concentrated in $\mathbb N$.

Thanks in advance,

Valerio