BDG inequalities for model-free continuous price paths with instant enforcement

Shafer and Vovk introduce in their book Game-theoretic foundations for probability and finance the notion of instant enforcement. In this paper we introduce an outer measure on the space of continuous paths which assigns zero value exactly to those sets (properties) of pairs of time $t$ and elementary event $omega$ which are instantly blockable. Next, for the introduced measure we prove BDG inequalities and use them to define It^o-type integral. Additionally, we prove few properties for the quadratic variation of model-free continuous paths which hold with instant enforcement.