On the Existence of Extreme Coherent Distributions with No Atoms

Abstract The paper is devoted to the study of extremal points of $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , the family of all bivariate coherent distributions on $$[0,1]^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . It is well-known that the set $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> is convex and $$\hbox {weak}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>weak</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> compact, and all extreme points of $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> must be supported on sets of Lebesgue measure zero. Conversely, examples of extreme coherent measures with a finite or countably infinite number of atoms have been successfully constructed in the literature. The main purpose of this article is to bridge the natural gap between those two results: We provide an example of extreme coherent distribution with an uncountable support and with no atoms. Our argument is based on classical tools and ideas from dynamical systems theory. This unexpected connection can be regarded as an independent contribution of the paper.