This paper is concerned with two aspects of the theory of measures on compact totally ordered spaces (the topology is to be the order topology). In Section 2, we clarify a recent construction of Sapounakis [11, 12] and, in so doing, we are able to say a little more about it. It should be added here that Sapounakis had other ends in view. To be precise, let I be the closed unit interval [0, 1] and let λ be Lebesgue measure on I. We shall construct another totally ordered set Ĩ which is compact in its order topology, a continuous increasing surjection τ : Ĩ → I with the property that card τ−1(t) = 2 for all t ∈ ]0,1[ (these brackets denote the open interval), and a measure
on Ĩ such that τ(
) = λ. Then the following theorem holds.