1 Sigma-Algebra: Describing measurable sets 6 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 12 1.1...

for any finite disjoint union ∪j=1Aj ×Bj of rectangles. It is easy to see, by considering a common refinement of any two representations of a set in A as a finite disjoint union of rectangles that m does not depend on the representation, and is well-defined premeasure on A. 1.2 DEFINITION (Product sigma-algebra). For any two measure spaces (X,M, μ) and (Y,N , μ), the product sigma algebra in X ...

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...

some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness.in this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by bernhard banaschewski and walt...

Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows tha...

In this paper we introduce a generalization of Meir-Keeler contraction forrandom mapping T : Ω×C → C, where C be a nonempty subset of a Banachspace X and (Ω,Σ) be a measurable space with being a sigma-algebra of sub-sets of. Also, we apply such type of random fixed point results to prove theexistence and unicity of a solution for an special random integral equation.

let $mathcal{a}$ be a banach algebra and $mathcal{m}$ be a banach $mathcal{a}$-bimodule. we say that a linear mapping $delta:mathcal{a} rightarrow mathcal{m}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{a} rightarrow mathcal{m}$ such that $delta(ab) = delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{a}$. giving some facts concerning general...

Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by Bernhard Banaschewski and W...

We consider Chern-Simons gauged nonlinear sigma model with boundary which has a manifest bulk diffeomorphism invariance. We find that the Gauss’s law can be solved explicitly when the nonlinear sigma model is defined on the Hermitian symmetric space, and the original bulk theory completely reduces to a boundary nonlinear sigma model with the target space of Hermitian symmetric space. We also st...