This paper studies vector-measure games in which the set of players is represented by a mixed measure space with atoms and with an atom-less part. The bargaining set introduced in Mas-Colell (J Math Econ 18(2):129–139, 1989) for a continuum exchange economy is adapted and analyzed within this (TU) market game. It is shown that the equal treatment imputations in the bargaining set have some interesting properties. In particular, we prove an extension of the budgetary exploitation property of core allocations in mixed markets to equal-treatment imputations in the Mas-Colell bargaining set.