  We define a measure of noncompactness $\lambda$ on the standard Hilbert
$C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that
$\lambda(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is
$\varepsilon$-close to a finitely generated projective submodule for any
$\varepsilon>0$) and derive its properties. Further, we consider the known,
Kuratowski, Hausdorff and Istr\u{a}\c{t}escu measure of noncomapctnes on
$l^2(\mathcal A)$ regarded as a locally convex space with respect to a suitable
topology, and obtain their properties as well as some relationship between them
and introduced measure of noncompactness $\lambda$.
