Possibilistic measures are usually defined as set functions ascribibg to each subset of the universe of cliscourse a real number from the unit interval and obeying sonie well-kiiown simple conditions. For the number of reasons, as a more realistic version of this model, let us consider partial possibilistic measures defined only for certain subsets and ascribing to them, instead of real numbers, elements from a more general structure. As a rule, a complete lattice will play this role, so let us pick up rather the qualitative and comparative than the quantitative features of particular degrees of possibility. Following the ideas of the standard measure theory, we define the inner and the outer measure induced by the partial latticevalued possibilistic measure in question. A subset of the basic universe is defined as ahnost measurable, if the difference (or rather distance) between the values of the inner and the outer measure ascribed to this set does not exceed, in the sense of the partial ordering relation defined in the used complete lattice, some given threshold value (a “small” fixed element from this lattice). Properties of systems of almost measurable sets are investigated in greater detail and some assertions related to them are introduced.