These are some brief notes on measure theory, concentrating on Lebesgue measure on Rn. Some missing topics I would have liked to have included had time permitted are: the change of variable formula for the Lebesgue integral on Rn; absolutely continuous functions and functions of bounded variation of a single variable and their connection with Lebesgue-Stieltjes measures on R; Radon measures on ...

We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N , there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncompact Riemannian symmetric spaces.

Let X be a topological space, and let C(X) be the complex of singular cochains on X with coefficients in R. We denote by C c (X) (resp. C ∗ b (X)) the subcomplex of C(X) given by continuous (resp. locally bounded Borelian) cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous (resp. locally bounded Borelian) rea...

$K$-frames as a generalization of frames were introduced by L. Gu{a}vruc{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new ge...

Our paper is devoted to two classical problems of category theory: the Orthogonal Subcategory Problem which asks, given a class H of morphisms, whether the subcategory H⊥ of all objects orthogonal to every member of H is reflective. And the Small Object Argument which asks whether the subcategory InjH of all objects injective to every member of H is weakly reflective and, moreover, the weak ref...

In this paper, a new definition of bounded fuzzy linear orderhomomorphism on $I$-topological vector spaces is introduced. Thisdefinition differs from the definition of Fang [The continuity offuzzy linear order-homomorphism. J. Fuzzy Math. {bf5}textbf{(4)}(1997), 829--838]. We show that the ``boundedness"and `` boundedness on each layer" of fuzzy linear orderhomomorphisms do not imply each other...

In this paper, we obtain the general solution and the generalized   Hyers-Ulam-Rassias stability in random normed spaces, in non-Archimedean spaces and also in $p$-Banach spaces and finally the stability via fixed point method for a functional equationbegin{align*}&D_f(x_{1},.., x_{m}):= sum^{m}_{k=2}(sum^{k}_{i_{1}=2}sum^{k+1}_{i_{2}=i_{1}+1}... sum^{m}_{i_{m-k+1}=i_{m-k}+1}) f(sum^{m}_{i=1, i...

sn-networks were first introduced by Lin [12], which are the concept between weak bases and cs-networks. sn-metrizable spaces [6] (i.e., spaces with σ-locally finite sn-networks) are one class of generalized metric spaces, and they play an important role in metrization theory, see [6, 13]. In this paper, we give a mapping theorem on sn-metrizable spaces, discuss relationships among spaces with ...

1. In the theory of (F)-spaces,1 the following two questions arise: (A) Let £ be a metrizable locally convex space; is it true that if every bounded set in E is separable, the space E is separable? (B) Let £ be a metrizable locally convex space, E its completion; is it true that every bounded set in E is contained in the closure of a bounded set of El The purpose of this note is to prove that, ...