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...