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A qualification is necessary: an arbitrary symbolic Gibbs measure does not satisfy the claim. Consider the doubling map with its two binary branches and a Bernoulli symbolic Gibbs measure with probabilities $2/3$ and $1/3$. This is the Gibbs law for a locally constant symbolic potential, while every level-$n$ interval has length $2^{-n}$. Along the nonboundary alternating itinerary, for even $n$,No finite $g$ can approximate these $g_n$ with a decaying uniform error. This proves the reusable fact that an arbitrary Gibbs measure need not be absolutely continuous.

For the intended geometric version, take the Gibbs measure for the geometric potential of an expanding map $-\log |T^{\prime }|$, with the standard sufficiently smooth finite full-branch hypotheses. The allowed transfer-operator result supplies a strictly positive Lipschitz continuous invariant probability density function $h$, normalized by $\int h=1$, and its symbolic weights of cylinder sets are $\nu (C_w)={\int }_{{\Delta }_w}h(x)dx$. State the needed regularity explicitly: $h\ge h_{\ast }>0$ and $|h(x)-h(y)|\le L_h|x-y|$. Expansion gives $\mathrm{diam}{\Delta }_w\le {\lambda }^{-n}$.

Let $\pi (ϵ)$ denote the point coded by the infinite sequence, and define $g(ϵ)=\log h(\pi (ϵ))$. The expression defining $g_n$ is the logarithm of the average of $h$ over the cylinder. For $x=\pi (ϵ)\in {\Delta }_w$,Since the logarithm is Lipschitz continuous with constant $1/h_{\ast }$ on this positive range, the cylinder averages of an invariant density giveThis proves the intended estimate. Expansion alone and an unspecified Gibbs symbolic potential are insufficient; the geometric choice and the regularity producing a positive Lipschitz continuous probability density function are substantive hypotheses.