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Moyennes limites et convexité. (Limit mean values and convexity). (French) Zbl 0673.26004
Let F be the set of bounded functions defined on \({\mathbb{R}}_+\), taking nonnegative real values, and Lebesgue-integrable on every interval [0,t]. To each function a in F we associate the ordered pair L(a) of the upper and lower limits of \(t^{-1}\int^{t}_{0}a(s)ds\), as t tends to \(+\infty\). We characterize the set \[ S(a)=\{L(b)\in {\mathbb{R}}^ 2;\quad b\in F\quad and\quad \forall t\in {\mathbb{R}}_+,\quad b(t)\leq a(t)\} \] as being a closed convex region of \({\mathbb{R}}^ 2\).
Reviewer: G.Grekos

MSC:
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
52A10 Convex sets in \(2\) dimensions (including convex curves)
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
28A25 Integration with respect to measures and other set functions
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