Let be a continuous map. CAO Yongluo proved that if is continous and for all , then

for EVERY , there exists such that the Birkhoff average ;

Moreover there exists a uniform such that for every and .

Now let be a family of continuous subadditive potential with respect to . By subadditive ergodic theorem we know that the limit exists on a set of full probability.

CAO: If on a set of full probability, then there exists a uniform and such that

for every .