  In this paper, we establish representation theorems for generators of
backward stochastic differential equations (BSDEs in short) in probability
spaces with general filtration from the perspective of transposition solutions
of BSDEs. As applications, we give a converse comparison theorem for generators
of BSDEs and also some characterizations to positive homogeneity, independence
of y, subadditivity and convexity of generators of BSDEs. Then, we extend
concepts of g-expectations and conditional g-expectations to the probability
spaces with general filtration and investigate their properties.
