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

We propose a set-indexed family of capacities { cap G } 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 ( cap R + ) 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 > 1 such that for all r > 0 , 1 a K G ( r 6 ) exp ( − π 2 8 r 2 ) ≤ cap G { f ⋆ ≤ r } ≤ a K G ( r 6 ) exp ( − π 2 8 r 2 ) . Here, K G denotes the Kolmogorov ε -entropy of G , and f ⋆ ≔ sup [ 0 , 1 ] | f | .