Derivation Theorems for σ-additive Set Functions under Assumptions of the Vitali Type

Abstract

The classical Vitali theorem on the real line R asserts that if ${\cal{V}}$ is a closed interval covering of a set X (mod ${{\cal{N}}^*}$) such that almost all points of X belong to intervals of ${\cal{V}}$ of arbitrarily small length, then for any ε > 0 there exists an enumerable (countable) disjoint subfamily {V _n } of ${\cal{V}}$ covering X (mod ${{\cal{N}}^*}$) and satisfying μ̄(S−S · X) < ε, where $S = \bigcup\limits_n {{V_n}}$ and μ denotes Borel measure on R.