Does there exist a probability finitely additive measure on $\mathbb N$ which is idempotent with respect to addition and multiplication simultaneously?
It is known (due to Hindman) that there is no ultrafilter that is additively and multiplicatively idempotent. But in the closure of the set of additively idempotent ultrafilters there are multiplicatively idempotent ultrafilters.
This question is related to the ones of Justin Moore: Idempotent measures on the free binary system? and Do distinct idempotent measures on finite binary systems have distinct supports?. There is good background in the first references, but I do repeat some definition...
Consider $l_\infty(\mathbb{N})$ with $\mathbb R$ as a base field. A finitely additive probability measure on $\mathbb{N}$ is $\mu\in l_\infty^*(\mathbb{N})$ which is positive ($\mu(f)\geq 0$ if $f\geq 0$) and $\mu(1)=1$. Denote the set of these measures as $PM(\mathbb{N})$. Let $\star=+$ or $\cdot$. We may define $\star$ on $\mu\in l_\infty^*(\mathbb{N})$ as follows: $$ \nu\star\mu(f)=\nu_x(\mu_y(f(x\star y))), $$ where $\mu_y(f(x\star y))$ means that we apply $\mu$ with respect to $y$ for $x$-shifts of $f$, the result is a function of $x$. It looks that the only algebraic property of $+,\cdots$ conserved by this extension is associativity. The set $PM(\mathbb{N})$ is closed with respect to $\star$. So, in this notation the question is
Does the exist $\mu\in PM(\mathbb{N})$ such that $\mu+\mu=\mu\cdot\mu=\mu$?
