Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as $$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$ where $\mu^{*n}$ is the $n$th convolution power of $\mu$. Usually, the Green function is defined for probability measures $\mu$, but it can be defined in more general settings, a typical example begin the measure $r\mu$, where $\mu$ is a probability measure and $r$ is a positive number below the spectral radius of $\mu$.
Of course, $G(\mu)$ can be infinite. We are interested in the following on measures $\mu$ such that $G(\mu)<+\infty$. To simplify things, we also restrict ourselves to finitely supported measures. Precisely, let $S$ be a finite set generating $G$ as a semi-group. Then, let $$P(S)^+=\{\mu \text{ finite measure whose support is exactly }S, G(\mu)<+\infty\}.$$ Endow the set of finite measures with the topology of pointwise convergence, that is $\mu_n$ converges to $\mu$ if and only if for every $g\in G$, $\mu_n(g)$ converges to $\mu(g)$.
Is the function $G(\mu)$ continuous in $\mu$ on $P(S)^+$ ?
