  With an emphasis on generators with quadratic growth in the control variable
we consider measure solutions of BSDE, a solution concept corresponding to the
notion of risk neutral measure in mathematical finance. In terms of measure
solutions, solving a BSDE reduces to martingale representation with respect to
an underlying filtration. Measure solutions related to measures equivalent to
the historical one provide classical solutions. We derive the existence of
measure solutions in scenarios in which the generating functions are just
continuous, of at most linear growth in the control variable (corresponding to
generators of at most quadratic growth in the usual sense), and with a random
bound in the time parameter whose stochastic integral is a BMO martingale. Our
main tools include a stability property of sequences of measure solutions, for
which a limiting solution is obtained by means of the weak convergence of
measures.
