Representation of filtration-consistent nonlinear expectation by g-expectation in general framework

In this paper, we mainly study nonlinear expectations on the general domain whose elements can be measurable with respect to a sigma-filed generated by the whole path of Brownian motion on infinite time interval. And we obtain that every translation invariant filtration consistent nonlinear expectation with some dominated condition can be represented by a g-expectation introduced by a backward stochastic differential equation on the general time horizon.