The McShane integral of functions defined on an -dimensional interval is considered in the paper. This integral is known to be equivalent to the Lebesgue integral for which the Vitali convergence theorem holds. For McShane integrable sequences of functions a convergence theorem based on the concept of equi-integrability is proved and it is shown that this theorem is equivalent to the Vitali convergence theorem.
Several mean value theorems for higher order divided differences and approximate Peano derivatives are proved.
For a differentiable function where is a real interval and , a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean such that are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.
Properties of metrically convex functions in normed spaces (of any dimension) are considered. The main result, Theorem 4.2, gives necessary and sufficient conditions for a function to be metrically convex, expressed in terms of the classical convexity theory.
Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s...
The existence of a minimal element in every equivalence class of pairs of bounded closed convex sets in a reflexive locally convex topological vector space is proved. An example of a non-reflexive Banach space with an equivalence class containing no minimal element is presented.