We establish the duality-formula for the superreplication price in a setting of volatility uncertainty which includes the example of "random $G$-expectation". In contrast to previous results, the contingent claim is not assumed to be quasi-continuous.
"Superreplication under volatility uncertainty for measurable claims." Electron. J. Probab. 18 1 - 14, 2013. https://doi.org/10.1214/EJP.v18-2358