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- W2022151931 abstract "Let f and g be continuous functions of bounded variation on [0, 1]. We use the Dini derivates of f and g to give a necessary and sufficient condition that the equation V(f + g) = V(f ) + V(g) holds. Let f and g be continuous functions of bounded variation on [0, 1]. We know that V(f + g) 0 (or max(D+f(x), D+f(x), D-f(x), D_f(x)) < 0). We offer the following THEOREM 1. Let Ef denote the set of points where f is increasing and g is decreasing and Eg denote the set of points where g is increasing and f is decreasing. Then a necessary and sufficient condition for (*) V( + g) = V() + V(g) to hold is that there exist sets Sf and Sg such that Ef U Eg = Sf U Sg and XfSf = XgSg = 0, where X is Lebesgue outer measure. Moreover, a necessary condition for (*) to hold is that X(Ef U Eg) = 0. The idea is that (*) holds when there are not too many points where one function increases and the other decreases. Before proving Theorem 1, let us discuss some of its consequences. If f is absolutely continuous and g is singular, then X(Ef U Eg) = 0 because g' = 0 almost everywhere. Then Xf(Ef U Eg) = 0 and (*) holds. Thus (*) might hold even when f is strictly increasing and g is strictly decreasing on [0, 1]. Just make one function absolutely continuous and the other singular. Of course (*) cannot hold if f is increasing and g is decreasing at each point of some subinterval of [0, 1]. Received by the editors June 10, 1981. 1980 Mathematics Subject Classification. Primary 26A45, 26A24, 26A30, 26A46." @default.
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- W2022151931 date "1982-04-01" @default.
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- W2022151931 title "When total variation is additive" @default.
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- W2022151931 doi "https://doi.org/10.1090/s0002-9939-1982-0643738-2" @default.
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