Based on the Halmos-Savage theorem, a new tractable criterion is confirmed on recognizing whether a family of probabilities can be dominated by a single probability or not, the implementation of which to the probability variety involved in $G$-expectation evinces the absence of a dominating probability. Therefore even though the static model of $G$-expectation can be constructed with the help of countably infinitely many probabilities those can be dominated simultaneously, when it comes to dynamic situation, such dominating probability measure does not exist. The key ingredient of our proof relies on the normability of Wasserstein metric of order one in probability space.

 

海报