  We develop in more detail our reweighting method for incorporating new
datasets in parton fits based on a Monte Carlo representation of PDFs. After
revisiting the derivation of the reweighting formula, we show how to construct
an unweighted PDF replica set which is statistically equivalent to a given
reweighted set. We then use reweighting followed by unweighting to test the
consistency of the method, specifically by verifying that results do not depend
on the order in which new data are included in the fit via reweighting. We
apply the reweighting method to study the impact of LHC W lepton asymmetry data
on the NNPDF2.1 set. We show how these data reduce the PDF uncertainties of
light quarks in the medium and small x region, providing the first solid
constraints on PDFs from LHC data.
