  Integral measures of weak gravitational lensing by dark-matter haloes, like
the aperture mass, are sensitive to different physical halo properties
dependent on the halo mass density profile. For isothermal profiles, the
relation between aperture mass and virial mass is steeper than for haloes with
the universal NFW profile. Consequently, the halo mass range probed by the
aperture mass is much wider for NFW than for isothermal haloes. We use recent
modifications to the Press-Schechter mass function in CDM models normalised to
the local abundance of rich clusters, to predict the properties of the halo
sample expected to be accessible with the aperture mass technique. While ~10
haloes should be detected per square degree if the haloes have NFW profiles,
their number density is lower by approximately an order of magnitude if they
have isothermal profiles. These results depend only very mildly on the
cosmological background model. We conclude that counts of haloes with a
significant weak-lensing signal are a powerful discriminator between different
dark-matter profiles.
