  In this paper, we study the behavior of the sets of volumes of the form
$\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a
nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we
focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index,
and $B$ has coefficients in a given DCC set. First, we show that such sets of
volumes satisfy the DCC property in the case of surfaces. Once this is
established, we show that surface pairs with given volume and for which
$K_X+B+M$ is ample form a log bounded family. These generalize results due to
Alexeev [Ale94].
