  The purpose of this paper is devoted to studying representation of measures
of non generalized compactness, in particular, measures of noncompactness, of
non-weak compactness, and of non-super weak compactness, etc, defined on Banach
spaces and its applications. With the aid of a three-time order preserving
embedding theorem, we show that for every Banach space $X$, there exist a
Banach function space $C(K)$ for some compact Hausdorff space $K$, and an
order-preserving affine mapping $\mathbb T$ from the super space $\mathscr B$
of all nonempty bounded subsets of $X$ endowed with the Hausdorff metric to the
positive cone $C(K)^+$ of $C(K)$ such that for every convex measure, in
particular, regular measure, homogeneous measure, sublinear measure of non
generalized compactness $\mu$ on $X$, there is a convex function $\digamma$ on
the cone $V=\mathbb T(\mathscr B)$ which is Lipschitzian on each bounded set of
$V$ such that \[\digamma(\mathbb T(B))=\mu(B),\;\;\forall\;B\in\mathscr B.\] As
its applications, we show a class of basic integral inequalities related to an
initial-value problem in Banach spaces, and prove a solvability result of the
initial-value problem, which is an extension of some classical results due to
Goebel, Rzymowski, and Bana\'{s}.
