A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^0$-simplicial subdivisions of $L^0$-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.