Capacities in Wiener space, quasi-sure lower functions, and Kolmogorov’s εε-entropy

We propose a set-indexed family of capacities {capG}G⊆R+ on the classical Wiener space C(R+). This family interpolates between the Wiener measure (cap{0}) on C(R+) and the standard capacity (capR+) on Wiener space. We then apply our capacities to characterize all quasi-sure lower functions in C(R+). In order to do this we derive the following capacity estimate which may be of independent interest: There exists a constant a>1a>1 such that for all r>0r>0, 1aKG(r6)exp(−π28r2)≤capG{f⋆≤r}≤aKG(r6)exp(−π28r2). Here, KG denotes the Kolmogorov εε-entropy of GG, and f⋆≔sup[0,1]|f|f⋆≔sup[0,1]|f|.