Given p(1,2], the wellposedness of backward stochastic differential equations with jumps (BSDEJs) in Lp sense gives rise to a so-called g-expectation with Lp domain under the jump filtration (the one generated by a Brownian motion and a Poisson random measure). In this paper, we extend such a g-expectation to a nonlinear expectation E with Lp domain that is consistent with the jump filtration. We study the basic (martingale) properties of the jump-filtration consistent nonlinear expectation E and show that under certain domination condition, the nonlinear expectation E can be represented by some g-expectation.