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.

A Note on the Product Borel Probability Measure

In this paper,a necessary and sufficient condition of a measure to be the product Borel probability measure on the product space of some compact metric spaces are given.

A Cost-benefit Measuring Model of Green Products in Manufacturing Industry

The paper builds up a cost-benefit measuring model of green products in manufacturing industry throughout its full life cycle, which can quantify green products＇ cost and benefit completely and correctly under the circumstance of satisfying enterprise, customer, environment and society. It also puts forth an operable method to estimate social benefit by opportunity cost and establishes a profit maximization-programming model. The model can be applied to justify whether some kinds of green products should be developed and produced.

On 3rd-Dimensional Product of Vertex Measurable Graphs

In this paper, first, a 3rd-dimensional vertex measurable graphs G is defined, which is an extension of the concept that was introduced in [3]. G = G1 × G2 × G3 is a graph defined over algebra ζ1 ×ζz × ζ3, which consists of all vertex sets that produce sub graphs of G. G1,G2, and G3 are three simple graphs, provided that （G1,ζ1）,（G2,ζz）, and （G3,ζ3） are three vertex measure spaces. Second, in order to maximize the edge＇s set, we present an alternative version of the definition of two-dimension Cartesian product of vertex measurable graphs that was given in [3], with preserving the same properties of the graphs and sub graphs that were illustrated.

Industrial Green Spatial Pattern Evolution of Yangtze River Economic Belt in China

We use the directional slacks-based measure of efficiency and inverse distance weighting method to analyze the spatial pattern evolution of the industrial green total factor productivity of 108 cities in the Yangtze River Economic Belt in 2003–2013.Results show that both the subprime mortgage crisis and ‘the new normal' had significant negative effects on productivity growth,leading to the different spatial patterns between 2003–2008 and 2009–2013.Before 2008,green poles had gathered around some capital cities and formed a tripartite pattern,which was a typical core-periphery pattern.Due to a combination of the polarization and the diffusion effects,capital cities became the growth poles and ‘core' regions,while surrounding areas became the ‘periphery'.This was mainly caused by the innate advantage of capital cities and ‘the rise of central China' strategy.After 2008,the tripartite pattern changed to a multi-poles pattern where green poles continuously and densely spread in the midstream and downstream areas.This is due to the regional difference in the leading effect of green poles.The leading effect of green poles in midstream and downstream areas has changed from polarization to diffusion,while the polarization effect still leads in the upstream area.

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.

Responses of Grassland Net Primary Productivity to Environmental Variables in Northern China

Various environmental factors affect net primary productivity （NPP） of grassland ecosystem. Extensive reports on the effects of environmental variables on NPP can be found in literature. However, the agreement on the relative importance of various factors in shaping the spatial pattern of grassland NPP has not yet been reached. Here a grassland in situ NPP database comprising 602 samples in northern China for 1980-1999 was developed based on a literature review of published biomass and forage yield field measurements. Correlation analyses and dominance analysis were used to quantify the separate and combined effects of environmental variables （climate topography and soil） on spatial variation in NPP separately. Grassland NPP ranged from 4.76 g C m-2a-1 to 975.94gCm-2a-1, showing significant variations in space. NPP increased with annual precipitation and declined with annual mean temperature significantly. Specifically, precipitation had the greatest impact on deserts, followed by steppes and meadows. Grassland NPP decreased with increasing altitude because of water limitation, and positively correlated with slope, but weakly correlated with aspect. Soil quality showed positive effects on NPP. Annual precipitation was the dominant factor affecting the spatial variability of net primary productivity, followed by elevation.