= Gibbs measure
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A <Gibbs measure> gives probabilities controlled by exponential weights from a potential. The precise convention must be specified: finite-volume conditional distributions and symbolic orbit-weight bounds are related formulations with different hypotheses.
For a finite-alphabet <full shift>, a standard symbolic Gibbs condition for a <probability measure> $\mu$ is that a constant $C\geq1$ satisfy
$$
C^{-1}\leq\frac{\mu([w])}{\exp(S_n\phi(x)-nP(\phi))}\leq C
$$
for every length-$n$ word $w$ and every $x$ in its <cylinder set> $[w]$. Here $S_n\phi(x)=\sum_{j=0}^{n-1}\phi(\sigma^jx)$, and $P(\phi)$ is <topological pressure>. The same constant must work for every word and every length. Summable variations make the orbit-weight comparison within one cylinder uniform. In applications to equilibrium states, shift invariance is also required.
For a one-symbol potential $\phi(x)=\log p_{x_0}$, where every $p_a>0$ and $\sum_ap_a=1$, the independent-symbol law has
$$
\mu([w_0\cdots w_{n-1}])=\prod_{j=0}^{n-1}p_{w_j},\qquad P(\phi)=0.
$$
The bound is exact with $C=1$. Equal symbol probabilities and unequal symbol probabilities are both Gibbs laws, although their geometric images can be very different: <an arbitrary Gibbs measure need not be absolutely continuous>.
An arbitrary unnormalized list of one-sided weights need not give consistent conditional probabilities for every finite horizon. Normalization and compatibility are part of constructing the measure, rather than consequences of calling the weights Gibbs.
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