284 6. ENTROPY AND NO LOCAL COLLAPSING8. Notes and commentary
Section 5. For T < oo the condition on rVtk just says that the rk are
uniformly bounded. In Definition 6.56, condition (1) may be replaced by(1') For some C < oo, IRm [g (tk)] I ::::; C r-;;^2 in B (Pk, rk).
We call the original definition the 'first definition' and the definition
with (1') replacing (1) the 'second definition'. If g (t) is locally collapsing
according to the first definition, then taking C = 1, it is locally collapsing
according to the second definition with the same rk. On the other hand,
if 9ij (t) is locally collapsing according to the second definition, then the
following hold.
(i) If C::::; 1, then g (t) is locally collapsing according to the first defi-
nition with the same rk.
(ii) If C 2: 1, then set rk ~ rk/VC ::::; rk. Then JRm [g (tk)]J ::::; r-;;^2
in B (Pk,rk), and we still have limk_, 00 r-;;nvolB (Pk,rk) = 0, so
that 9ij (t) is locally collapsing according to the first definition with
rk = rk/VC.