We study a fuzzy type integral for measurable multifunctions with respect to a fuzzy measure. Some classical properties and convergence theorems are presented.

We generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

In this paper we show that the category of measurable spaces is closed under coproducts in the category of sets. For an arbitrary ring R, we define measurable and probability right R-modules and we prove that the categories of these new objects are closed under kernels, cokernels and pushouts in the category of right R-modules. We also show that the category of measurable right R-modules is closed under coproducts and products in the category of right R-modules. We end this paper by giving some results about stochastically independence in the category of probability right R-modules.

Despite extensive research carried out on sludge bulking, it is still a common problem world-wide and a comprehensive solution is yet to be developed. In this research, the effect of three-stage anoxic selector on controlling activated sludge bulking is investigated in laboratory sludge systems. These experiments were carried out on two pilot plants; a three-stage anoxic selector preceding a complete mix system (experimental unit) and a complete mix system (control unit). Results from this study indicate that in selector zones with F/M ratios high enough to allow for rapid substrate uptake to occur predominately by floc forming microorganisms, the sludge settleability is influenced by the mode of substrate removal in the selector and by floc loading along the following lines. The substrate storage mechanism predominates in substrate removal for lower floc loadings, leading to improved sludge settleability, while for higher floc loadings the high rate metabolic mechanism predominates for substrate removal, leading to impoverished sludge settleability.

We study Beck-like coloring of measurable functions on a measure space Ω taking values in a measurable semigroup Δ . To any measure space Ω and any measurable semigroup Δ , we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Ω and having values in Δ , with two vertices f and g adjacent if f
g = 0 a. e. . We show that, if Ω is atomic, then not only the Beck’ s conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph.