  $L\sp{p}$ space is a crucial aspect of classical measure theory. For
nonadditive measure, it is known that $L\sp{p}$ space theory holds for the
Choquet integral whenever the monotone measure $\mu$ is submodular and
continuous from below. The main purpose of this paper is to generalize
$L\sp{p}$ space theory to $+,\cdot$-based pan-integral. Let $(X, {\cal A},
\mu)$ be a monotone measure space. We prove that the $+,\cdot$-based
pan-integral is additive with respect to integrands if $\mu$ is subadditive.
Then we introduce the pan-integral for real-valued functions(not necessarily
nonnegative), and prove that this integral possesses linearity if $\mu$ is
subadditive. By using the linearity of pan-integral, we finally show that all
of the pan-integrable functions form a Banach space. Since the $+,\cdot$-based
pan-integral coincides with the concave integral for subadditive measure, the
results obtained in this paper remain valid for the concave integral. Noticing
that an outer measure is subadditive, we can define a Lebesgue-like
integral(possesses linearity) from an outer measure, and the $L\sp{p}$ theory
holds for this integral.
  {\it Keywords:} Monotone measure; Subadditivity; Pan-integral; Linearity;
Pan-integrable space; Completeness
