Given $$p \in (1,2]$$ p ∈ ( 1 , 2 ] , the wellposedness of backward stochastic differential equations with jumps (BSDEJs) in $$\mathbb {L}^p$$ L p sense gives rise to a so-called g-expectation with $$\mathbb {L}^p$$ L p 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 $$\mathcal{E}$$ E with $$\mathbb {L}^p$$ L p domain that is consistent with the jump filtration. We study the basic (martingale) properties of the jump-filtration consistent nonlinear expectation $$\mathcal{E}$$ E and show that under certain domination condition, the nonlinear expectation $$\mathcal{E}$$ E can be represented by some g-expectation.