The introduction of the notion of independence in possibility theory i
s a problem of long-standing interest. Many of the measure-theoretic d
efinitions that have up to now been given in the literature face some
difficulties as far as interpretation is concerned. Also, there are in
consistencies between the definition of independence of measurable set
s and possibilistic variables. After a discussion of these definitions
and their shortcomings, a new measure-theoretic definition is suggest
ed, which is consistent in this respect, and which is a formal counter
part of the definition of stochastic independence in probability theor
y. In discussing the properties of possibilistic independence, I draw
from the measure- and integral-theoretic treatment of possibility theo
ry, discussed in Part I of this series of three papers. I also investi
gate the relationship between this definition of possibilistic indepen
dence and the definition of conditional possibility, discussed in deta
il in Part II of this series. Furthermore, I show that in the special
case of classical, two-valued possibility the definition given here ha
s a straightforward and natural interpretation.