The aim of this note is to study the submajorization inequalities for $tau$-measurable operators in a semi-finite von Neumann algebra on a Hilbert space with a normal faithful semi-finite trace $tau$. The submajorization inequalities generalize some results due to Zhang, Furuichi and Lin, etc..

This paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

Laczkovich proved that if bounded subsets A and B of R have the same non-zero Lebesgue measure and the upper box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, thi...

We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is א0. We also show that if add(null) = c, then the universally measurable chromatic number of every treei...

Background Eliciting and specifying customer requirements in a precise and unambiguous way is critical to the success of a project. However anyone who has done any requirements engineering also knows that it is a very difficult activity involving many diverse skills. An important reason for the degree of difficulty is that requirements engineering involves many different people. Each person has...