Vol. 43, No. 2, 1972

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Weighted convergence in length

William Richard Derrick

Vol. 43 (1972), No. 2, 307–315

This paper studies the lower semicontinuity of weighted length

           ∫        ∫
(∗)   lim inf   fds ≧   fds,
n→ ∞    γn       γ

where the sequence of curves {γn} converges uniformly to the curve γ, and f is a nonnegative lower semicontinuous function. Necessary and sufficient conditions for equality in () are obtained, as well as conditions which prevent γ from being rectifiable. Requirements are given for the attainment of the weighted distance, from a point to a set, and the families of functions, for which weighted distance is attained or () is satisfied, are shown to be monotone closed from below. Finally, the solutions to the integral inequality

(∗∗)  |γ(t)− γ(0)| ≧     f ds,

are shown to be compact if the initial values γ(0) lie in a compact set.

Mathematical Subject Classification 2000
Primary: 28A75
Received: 10 May 1971
Revised: 20 July 1971
Published: 1 November 1972
William Richard Derrick