Lebesgue measure

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = [a,b], or I = (a, b), in the set \mathbb{R} of real numbers, let \ell(I)= b - a denote its length. For any

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A finitely presented infinite simple group of homeomorphisms of the circle

Yash Lodha

We construct a finitely presented, infinite, simple group that acts by homeomorphisms on the circle, but does not admit a non-trivial action by C1-diffeomorphisms on the circle. This is the first such example. The group emerges as a group of piecewise projective homeomorphisms of S1=R?{infinity}. We also show that it does not admit a non-trivial action by piecewise linear homeomorphisms of the circle. Another interesting and new feature of this example is that it produces a non-amenable orbit equivalence relation with respect to the Lebesgue measure.

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Coherent actions by homeomorphisms on the real line or an interval

Yash Lodha

We study actions of groups by orientation preserving homeomorphisms on R (or an interval) that are minimal, have solvable germs at +/-infinity and contain a pair of elements of a certain dynamical type. We call such actions coherent. We establish that such an action is rigid, i.e., any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non-elementary amenable, but satisfies that every proper quotient is solvable. The structure theory we develop allows us to prove a plethora of non-embeddability statements concerning groups of piecewise linear and piecewise projective homeomorphisms. For instance, we demonstrate that any coherent group action G < Horneo(+) (R) that produces a nonamenable equivalence relation with respect to the Lebesgue measure satisfies that the underlying group does not embed into Thompson's group F. This includes all known examples of nonamenable groups that do not contain non abelian free subgroups and act faithfully on the real line by homeomorphisms. We also establish that the Brown-Stein-Thompson groups F(2, pi, horizontal ellipsis ,p(n)) for n >= 1 and p(1), horizontal ellipsis ,p(n) distinct odd primes, do not embed into Thompson's group F. This answers a question recently raised by C. Bleak, M. Brin and J. Moore.

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