Representation theorems for generators of backward stochastic differential equations and their applications

We prove that the generator g   of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs at point (t,y,z)(t,y,z) if and only if t is a conditional Lebesgue point of generator g   with parameters (y,z)(y,z). By this conclusion, we prove that, if g is a Lebesgue generator and g is independent of y, then, Jensen's inequality for g-expectation holds if and only if g is super homogeneous; we also obtain a converse comparison theorem for deterministic generators of BSDEs.