Consider the set $S=\{2,3,\ldots\}$ equipped with the operation $n\cdot m=n^m$. 



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> **Question:** Do there exist a mean on $S$ which is left and right invariant with respect to $\cdot$?
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Thanks in advance,


Valerio



Mean on the natural numbers which is invariant with respect to the power
Posting it as an answer, since OP confirmed I did understand the question right.


Such a mean cannot exist. In fact, if it would, we could denote it by $M$ and compute:


$M\left(S\right)=M\left(2^S\right)+M\left(3^S\right)+M\left(S\setminus 2^S\setminus 3^S\right)$ (since the sets $2^S$, $3^S$ and $S\setminus 2^S\setminus 3^S$ are pairwise disjoint and cover $S$)


$\geq M\left(2^S\right)+M\left(3^S\right)$ (since the mean is positive)


$=M\left(S\right)+M\left(S\right)$ (since the mean is left-invariant, so that $M\left(p^S\right)=M\left(S\right)$ for every $p\in\mathbb S$)


and thus $0\geq M\left(S\right)$, contradiction.


We only used the left-invariance of $M$.


