  Given an almost additive sequence of continuous functions with bounded
variation $\mathcal{F}=\{\log f_n\}_{n=1}^{\infty}$ on a subshift $X$ over
finitely many symbols, we study properties of a function $f$ on $X$ such that
$\lim_{n\to\infty}\frac{1}{n}\int \log f_n d\mu=\int f d\mu$ for every
invariant measure $\mu$ on $X$. Under some conditions we construct a function
$f$ on $X$ and study a relation between the property of $\mathcal{F}$ and some
particular types of $f$. We also study the case when $\mathcal{F}$ is weakly
almost additive. As applications we study images of Gibbs measures for
continuous functions under one-block factor maps. In order to study conditions
for the image to be a Gibbs measure for a continuous function, we investigate a
relation between the almost additivity of the sequences associated to relative
pressure functions and the fiber-mixing property of a factor map.
