  We study various measures of weak lensing distortions in future surveys,
taking into account the noise arising from the finite survey size and the
intrinsic ellipticity of galaxies. We also consider a realistic redshift
distribution of the sources, as expected for the SNAP mission. We focus on the
low order moments and the full distribution function (pdf) of the aperture-mass
$\Map$ and of the smoothed shear component $\gammais$. We also propose new
unbiased estimators for low-order cumulants which have less scatter than the
usual estimators of non-Gaussianity based on the moments themselves. Then,
using an analytical model which has already been seen to provide a good
description of weak gravitational lensing through comparison against numerical
simulations, we study the statistical measures which can be extracted from
future surveys like the SNAP experiment. We recover the fact that at small
angular scales ($1'<\theta_s<10'$) the variance can be extracted with a few
percent level accuracy. Non-Gaussianity can also be measured from the skewness
of the aperture-mass (at a 10% level) while the shear kurtosis is more noisy
and cannot be easily measured beyond 6'. On the other hand, we find that the
pdf of the estimator associated with the aperture-mass can be distinguished
both from the Gaussian and the Edgeworth expansion and could provide useful
constraints, while this appears to be difficult to realize with the shear
component. Finally, we investigate various survey strategies and the
possibility to perform a redshift binning of the sample.
