  The paper introduces the concept of quasimeasure of noncompactness. Motivated
by the Arzel\`a-Ascoli theorem for $C^b(X,E)$, where $X$ is an Euclidean space
and $E$ an arbitrary Banach space, we construct a quasimeasure for this space
and study its properties. An analogon for Darbo fixed point theorem is obtained
with the additional aid of measure of nonconvexity. The paper ends with
possible application in integral equations. We prove that a Hammerstein
operator with Carath\'{e}odory kernel and nonlinearity of a certain type has a
fixed point.
