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Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsøe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.

It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

If L is a set of disjoint closed line segments in En, let E(L) denote the end set of L, i.e. the set of end points of members of L.

In [1, 2] V. L. Klee and M. Martin proved the following lemma:

IfLis a disjoint set of closed line segments in E2 such that E(L) is compact, then E(L) has zero 2-measure.

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Measures not approximable or not specifiable by means of balls

A compact set of disjoint line segments in E3 whose end set has positive measure

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342 results in 28Axx

Measures not approximable or not specifiable by means of balls

A compact set of disjoint line segments in E3 whose end set has positive measure