Borel measurability of extreme path derivatives (Q1094545)

The author deals with the Borel measurability of extreme path derivatives of real functions under some conditions concerning path systems and functions. The author considers a system of paths as a multifunction and its range as a metric space with the Hausdorff metric. At first, he proves that any extreme path derivative of a continuous real function of a real variable relative to a continuous system of paths is a function of Baire class two. Under some special added conditions, the extreme path derivatives of a real function of the first class of Baire are in the Baire class four. Then the author gives a function f of the second Baire class whose extreme path derivatives \(\bar f{}_ E'\) are not Borel measurable. There is also proved that any extreme path derivative of a Borel measurable function with respect to a continuous system of paths is Lebesgue measurable. The paper ends with a brief discussion on Borel measurability of path derivatives and the author proves that the path derivative of a function of the Baire class \(\alpha\) is in the Baire class \(\alpha +2\), when the system of paths is continuous.