Is there a fixed-point theorem that implies the following result?



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> Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to left shifts and closed with respect to the pointwise convergence. Then $F$ contains a constant function.
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This statement looks like [Ryll-Nardzewski fixed point theorem](https://en.wikipedia.org/wiki/Ryll-Nardzewski_fixed-point_theorem), but it does not seem to follow from the theorem.



A variation of the Ryll-Nardzewski fixed point theorem
The claim does not hold. Let $F$ be the free group on $\{a,b\}$ and $X⊂F$ be the words whose last letter is $a$ or $b$. Let $\xi=\delta\_a+\delta\_b−\delta\_{a^{-1}}−\delta\_{b^{-1}}\in\ell\_1(F)$. Then for any $s\in F$, one has $\langle 1\_X,s\xi\rangle\geq1$. (To see this, view $F$ as the $4$-regular tree and $s\xi$ as a signed characteristic function of the $1$-neighborhood of $s\in F$.) Hence the pointwise-topology closed convex hull of $F1\_X$ in $\ell\_\infty(F)$ cannot contain constant functions.


