Mathematical Expectation of Random Structure Spaces and Applications

Let (E, S, Ω, f) be an random structure space, when (E,S,Ω,f) is random metric space, then random metric is random variable. The mathematical expectation of random metric is quasi-metric. When (E,S,Ω,f) is random normed space, then random norm is random variable. The mathematical expectation of random norm is quasi-norm. When (E,S,Ω,f) is random inner product space, then random inner product is random variable. The mathematical expectation of random inner product is inner product. Further, the Riesz theorem of contiuous linear functional is proved by using mathematical expectation of random inner product.