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Some Properties of Variational Measures

Recently several authors have established a remarkable property of the variational measures associated with a function. Expressed in classical language, this property asserts that if a function is ACG∗ on all sets of Lebesgue measure zero then the function must be globally ACG∗. This article is an exposition of some ideas related to this property with the intention of bringing it to the attention of a wider audience than these original papers might attract. Recent years have seen continued interest in the variational measures associated with a function, e.g., [1], [2], [3], [4], [7], [8], [9], [12], [13], [14], [15], [17], [18], [20], and [21]. In the simplest setting a function f : [a, b]→ R is given and one constructs a measure μf that carries the variational information about f . If f is of bounded variation then μf is the usual Lebesgue-Stieltjes measure associated with the total variation function of f . In general a measure μf can be constructed for arbitrary functions and which has considerable power to express properties of f . Perhaps the nicest elementary uses of this measure would be in the following assertions. If f : [a, b] → R then a necessary and sufficient condition for the identity f(x) − f(a) = ∫ x a f ′(t) dt in the sense of the Lebesgue integral is that μf is finite and absolutely continuous with respect to Lebesgue measure on [a, b]. If f : [a, b] → R then a necessary and sufficient condition for the identity f(x)−f(a) = ∫ x a f ′(t) dt in the sense of the Denjoy-Perron integral is that μf is σ–finite and absolutely continuous with respect to Lebesgue measure on [a, b]. Mathematical Reviews subject classification: 26A45, 26A39, 28A12 Received by the editors August 20, 1998