  We describe a measurement of the time-integrated luminosity of the data
collected by the BABAR experiment at the PEP-II asymmetric-energy e+e- collider
at the Upsilon(4S), Upsilon(3S) and Upsilon(2S) resonances and in a continuum
region below each resonance. We measure the time-integrated luminosity by
counting e+e- --> e+e- and (for the Upsilon(4S) only) e+e- --> mu+mu- candidate
events, allowing additional photons in the final state. We use data-corrected
simulation to determine the cross sections and reconstruction efficiencies for
these processes, as well as the major backgrounds. Due to the large cross
sections of e+e- --> e+e- and e+e- --> mu+mu-, the statistical uncertainties of
the measurement are substantially smaller than the systematic uncertainties.
The dominant systematic uncertainties are due to observed differences between
data and simulation, as well as uncertainties on the cross sections. For data
collected on the Upsilon(3S) and Upsilon(2S) resonances, an additional
uncertainty arises due to Upsilon --> e+e-X background. For data collected off
the Upsilon resonances, we estimate an additional uncertainty due to
time-dependent efficiency variations, which can affect the short off-resonance
runs. The relative uncertainties on the luminosities of the on-resonance
(off-resonance) samples are 0.43% (0.43%) for the Upsilon(4S), 0.58% (0.72%)
for the Upsilon(3S), and 0.68% (0.88%) for the Upsilon(2S).
