  For a nonadditive measure $\mu$, the space $\mathcal{L}^0(\mu)$ of all
measurable functions, the Choquet-Lorentz space $\mathcal{L}^{p,q}(\mu)$, the
Lorentz space of weak type $\mathcal{L}^{p,\infty}(\mu)$, the space
$\mathcal{L}^\infty(\mu)$ of all $\mu$-essentially bounded measurable
functions, and their quotient spaces are defined together with suitable
prenorms on them. Among those function spaces, the Choquet-Lorentz space is
defined by the Choquet integral, while the Lorentz space of weak type is
defined by the Shilkret integral. Then the completeness and separability of
those spaces are investigated. A new characteristic of nonadditive measures,
called property (C), is introduced to establish the Cauchy criterion for
convergence in $\mu$-measure of measurable functions. This criterion and
suitable convergence theorems of the Choquet and Shilkret integrals provide
instruments for carrying out our investigation.
