**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Invariant measure

Summary

In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping.
Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
Definition
Let (X, \Sigma) be a measurable space and let f : X \to X be a measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be invariant under f if, for every measurable set A in \Sigma,
\mu\left(f^{-1}(A)\right) = \mu(A).
In terms of the pushforward measure, this states that f_*(\mu) = \mu.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (2)

Related courses (5)

PHYS-702: Advanced Quantum Field Theory

The course builds on the two previous courses on the subject. The main subject is the study of quantum field theories at the loop level. The course introduces the concept of loop divergences and renormalization. Non abelian gauge theories are also discussed in depth.

MATH-487: Topics in stochastic analysis

This course offers an introduction to Markov processes, a widely used model for random evolutions with no memory.

MATH-494: Topics in arithmetic geometry

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applications, notably we will prove rationality of the Weil Zeta function.

Related concepts (10)

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical mode

Ergodic theory

Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to pro

Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in

Related units

No results

Related publications (13)

Loading

Loading

Loading

Related lectures (4)

Gaussian measures μβ,ν are associated to some stochastic 2D models of turbulence.They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν.We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μβ,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μβ,ν-stationary solution (for any ν >0).

We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations, independently of the weak order of accuracy of the method. We then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method. Numerical experiments confirm our theoretical findings

Julian Kellerhals, Nicolas Monod

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product C*-algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups.