The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces

Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.

On Ordinary and Standard Products of Infinite Family of σ-finite Measures and Some of Their Applications

We introduce notions of ordinary and standard products of a-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on R∞ and Rogers-Fremlin measures on l∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal （i.e., 2c）. We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.

FUBINI THEOREM w.r.t. STOCHASTIC MEASURE ON PRODUCT MEASURABLE SPACE

Using the new results about the existence of product S.M.[1], we get two forms of Fubini theorem about product S.M. on product measurable space in § 1-§ 2. On being restricted to the special case of S.M. (I), the conditions needed are much weaker than those of [2] and couldn't be improved anymore. In the rest of this paper, we discuss how to calculate double integration w.r.t .non-product type S.M. on product space by iterated integration. Even in the casc of classical measure theory, the problem hasn't been thoroughly solved yet.Por the first two sections we suppose that (X, X), (Y, y), are measurable spaces,any two of which form a 'nice pair'[1], P is a probability mea.sure on is the P-completion of so is a complete probability space. Let L be the coniplete topological linear space which consists of all a.s. finite r.v. on (we identify those r.v. which differ only on a set of probability 0), If Z,W are valued S.M. on X, y respectively, then there uniquely exists an valued S.M. on X × y, denoted by Z × W, such that Z ×W(E ×F) = Z(E)W(F)for any E ∈ X, F ∈ y[1] . Thus we may discuss the double integrals of the X × y measurable fonction f = j(x,y) w.r.t. Z × W, denoted br (or shortly by , at least for thcoe f either ounded or nonnegative. we call f integrable w.r.t.Z ×W if both f+dZ × W and f-dZ×W ∈ so All integrable f form a complete topological linear space, denoted by (or shortly by L1(dZ × W))[3].In this paper, we discuss how to calculate the double stochastic integrals by iterated stochas tic integrals. Since there are two different ordare to calculate the iterated stochastic integrals,and both are equal to the sanie double stochastic integral, so tbe order of the iterated stochastic Received July 6, 1991. Revised January 29, 1993.This project is supported by the National Natural Sciences Foundation of China.integrals is exchangeable. In classical analysis, such a kind of statement is usually called the Fubini theorem.

The 10 Measures to Boost Grain Production

Enlarge the subsidy scope for winter wheat irrigation, keeping the subsidy standard at 150 yuan ($22.8) per hectare. Grant joint subsidies for winter wheat of 150 yuan ($22.8) per hectare.