On the possible values of upper and lower derivatives with respect to convex differential bases. (Q556595)

A basis is called convex if it consists of convex sets, it is called density-like if it differentiates integrals of characteristic functions of measurable sets, it is called centered if there exists an \(\varepsilon> 0\) such that for each \(x\in\mathbb{R}^n\) and \(E\in B(x)\) there exists a translation \(T\) for which \(T(E)\in B(x)\) and the coefficient of centeredness of \(T(E)\) at \(x\) is greater than or equal to \(\varepsilon\), the terms translation- and homothety-invariant are self-explaining. Theorem 1 says that if a convex density-like basis \(B\) in \(\mathbb{R}^n\) is translation- and homothety-invariant and centered, then both sets \(\{-\infty<\underline D_B(f,\cdot)< f\}\) and \(\{f<\overline D_B(f,\cdot)<\infty\}\) have measure zero for any nonnegative \(f\in L(\mathbb{R}^n)\).