With an additive function φ from a Boolean ring A into a normed space two positive functions on A, called semivariations of φ, are associated. We characterize those functions as submeasures with some additional properties in the general case as well as in the cases where φ is bounded or exhaustive.
Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.
It is shown that if A ⊂ ℝ has the same constant shade with respect to all Banach measures, then the same is true of any similarity transformation of A and the shade is not changed by the transformation. On the other hand, if A ⊂ ℝ has constant μ-shade with respect to some fixed Banach measure μ, then the same need not be true of a similarity transformation of A with respect to μ. But even if it is, the μ-shade might be changed by the transformation. To prove such a μ exists, a Hamel basis with some...
Consider a semigroup action on a set. We derive conditions, in terms of the induced action of the semigroup on {0,1}-valued probability charges, which ensure that all invariant probability charges are strongly continuous.
Let be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure is called -Zygmund if there exists a positive constant such that for any pair of adjacent cubes of the same size. Similarly, is called an - symmetric measure if there exists a positive constant such that for any pair of adjacent cubes of the same size, . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...