A physical measure is an invariant measure whose long-term statistical averages describe a set of initial
points of positive reference volume. More precisely, let
be a continuous
map of a compact metric space with a specified
finite reference Borel measure 
. An invariant Borel measure 
that is a probability
measure is physical relative to 
if the set of points 
satisfying
 |
for every continuous function 
has positive 
-measure (Young 2002). The usual
reference is ambient Lebesgue measure, or volume
coming from a Riemannian metric. The reference
measure is part of the definition.
The reference measure matters: a set of full 
-measure can have zero 
-measure. On a compact interval,
an ergodic measure with a probability
density function relative to Lebesgue measure
is physical. Indeed, apply Birkhoff's ergodic
theorem to a countable set of continuous
functions that is dense in the uniform norm.
Uniform approximation extends convergence to every continuous
function on a common set of full 
-measure. This set
must have positive Lebesgue measure. A physical
measure need not have a probability density
function.
