Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\longmapsto A$ for $T$ {\it disperses} if $\lim_{g\to\infty}c(g,\cdot )-\alpha (g)=\infty $ in measure for every map $\alpha \colon G\longrightarrow A$. We show that a cocycle $c$ for $T$ does not disperse if and only if there exists a compact subgroup $A_0\subset A$ such that the quotient cocycle $\bar{c}\colon G\times X\longrightarrow A/A_0$ of $c$ is {\it trivial} (i.e. cohomologous to a homomorphism $\eta \colon G\longrightarrow A/A_0$).
This result extends a number of earlier characterizations of coboundaries and trivial cocycles by tightness conditions on the distributions of the maps $\{c(g,\cdot ):g\in G\}$ and has the following consequence: let $T$ be a measure-preserving ergodic automorphism of a probability space $(X,\mu )$, $f$ a positive real-valued Borel map on $X$ with $\int f\,d\mu =1$, and let $T^f$ be the flow under the function $f$ with base $T$. If $T$ is mixing and $T^f$ is weakly mixing, or if $T$ is ergodic and $T^f$ is mixing, then the cocycle defined by $f$ disperses. Special cases of this result are due to Kocergin (1972), Khanin and Sinai (1992) and Lemanczyk (2000).