  We introduce non-linear traces of the Choquet type and Sugeno type on a
semifinite factor $\mathcal{M}$ as a non-commutative analog of the Choquet
integral and Sugeno integral for non-additive measures. We need weighted
dimension function $p \mapsto \alpha(\tau(p))$ for projections $p \in
\mathcal{M}$, which is an analog of a monotone measure. They have certain
partial additivities. We show that these partial additivities characterize
non-linear traces of both the Choquet type and Sugeno type respectively. Based
on the notion of generalized eigenvalues and singular values, we show that
non-linear traces of the Choquet type are closely related to the Lorentz
function spaces and the Lorentz operator spaces if the weight functions
$\alpha$ are concave. For the algebras of compact operators and factors of type
${\rm II}$, we completely determine the condition that the associated weighted
$L^p$-spaces for the non-linear traces become quasi-normed spaces in terms of
the weight functions $\alpha$ for any $0 < p < \infty$. We also show that any
non-linear trace of the Sugeno type gives a certain metric on the factor. This
is an attempt at non-linear and non-commutative integration theory on
semifinite factors.
