  We study non-linear traces of Choquet type and Sugeno type on the algebra of
compact operators. They have certain partial additivities. We show that these
partial additivities characterize non-linear traces of both Choquet type and
Sugeno type respectively. There exists a close relation between non-linear
traces of Choquet type and majorization theory. We study trace class operators
for non-linear traces of Choquet type. More generally we discuss Schatten-von
Neumann $p$-class operators for non-linear traces of Choquet type. We determine
when they form Banach spaces. This is an attempt of non-commutative integration
theory for non-linear traces of Choquet type on the algebra of compact
operators. We also consider the triangle inequality for non-linear traces of
Sugeno type.
