We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For so...

In his addendum to Saks, Theory of the Integral, Banach considers a Lebesgue integral defined in a manner quite similar to that of Daniell and remarks that no use is made of a measure. I t is, however, quite easy to show that Banach's and Daniell's integrals are expressible as Lebesgue integrals whose measure functions are regular outer measures in the sense of Carathéodory. In the first two se...

Since the 1930’s many authors have studied the distribution νλ of the random series Yλ = ∑±λn where the signs are chosen independently with probability (1/2, 1/2) and 0 < λ < 1. Solomyak recently proved that for almost every λ ∈ [ 1 2 , 1], the distribution νλ is absolutely continuous with respect to Lebesgue measure. In this paper we prove that νλ is even equivalent to Lebesgue measure for alm...