are within the distance of at least one of those chosen. Thus we are covering
the space to within
apart from a set of small measure . We define the dimension rate
for the ensemble
by the triple limit
log N
T
Lim Lim Lim
;
;
=
:
0
0 T
∞
T log
!
!
!
This is a generalization of the measure type definitions of dimension in topology, and agrees with the intu-
itive dimension rate for simple ensembles where the desired result is obvious.
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