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In this paper, we first use PDE techniques and probabilistic methods toidentify a kind of quasi-continuous random variables. Then we give acharacterization of the $G$-integrable processes and get a kind ofquasi-continuous processes by Krylov's estimates. This result is useful for thedevelopment of $G$-stochastic analysis theory. Moreover, it also provides atool for the study of the non-Markovian It\^o processes.

Quasi-continuous random variables and processes under the G-expectation framework