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

Shafer and Vovk introduce in their book \cite{ShaferVovk:2018} the notion of \emph{instant enforcement} and \emph{instantly blockable} properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of ''typical'' price paths. In this paper we introduce an outer measure on the space $[0, +\ns) \times \Omega$ which assigns zero value exactly to those sets (properties) of pairs of time $t$ and an elementary event $\omega$ which are instantly blockable. Next, for a slightly modified measure, we prove It\^o's isometry and BDG inequalities, and then use them to define an It\^o-type integral. Additionally, we prove few properties for the quadratic variation of model-free, continuous martingales, which hold with instant enforcement.