Projective curves over a field $K$ and arithmetic curves in Arakelov geometry have similar properties. For example, they are “compact” and have the “product formula”. Recently, Chen and Moriwaki introduced the notion of “adelic curves” to treat them in the same framework and study Arakelov geometry over an adelic curve. A (proper) adelic curve is a triplet $(K,\Omega,P)$ where $K$ is a field, $\Omega$ is a measure space parameterized by absolute values of $K$ and $P$ is a product formula. Among adelic curves, we focus on a trivially valued field, which is a field equipped with the trivial absolute value. In this talk, we study Arakelov theory over a trivially valued field and mainly discuss the bigness of adelic Cartier divisors. We show several properties of the volume function of an adelic Cartier divisor, for example, the continuity, the log-concavity and so on.