In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Let be a locally compact Hausdorff topological group. The -algebra generated by all open subsets of is called the Borel algebra. An element of the Borel algebra is called a Borel set. If is an element of and is a subset of , then we define the left and right translates of by g as follows: Left translate: Right translate: Left and right translates map Borel sets onto Borel sets. A measure on the Borel subsets of is called left-translation-invariant if for all Borel subsets and all one has A measure on the Borel subsets of is called right-translation-invariant if for all Borel subsets and all one has There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure on the Borel subsets of satisfying the following properties: The measure is left-translation-invariant: for every and all Borel sets . The measure is finite on every compact set: for all compact . The measure is outer regular on Borel sets : The measure is inner regular on open sets : Such a measure on is called a left Haar measure. It can be shown as a consequence of the above properties that for every non-empty open subset . In particular, if is compact then is finite and positive, so we can uniquely specify a left Haar measure on by adding the normalization condition . In complete analogy, one can also prove the existence and uniqueness of a right Haar measure on . The two measures need not coincide. Some authors define a Haar measure on Baire sets rather than Borel sets.

About this result
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 courses (9)
MATH-315: Spaces of non-positive curvature and groups
Non-positive curvature is a fundamental aspect of geometry appearing in Euclidean spaces, hyperbolic spaces, trees, buildings and many more spaces. We study it with the general but powerful tool of CA
MATH-338: Topological groups
We study topological groups. Particular attention is devoted to compact and locally compact groups.
MATH-432: Probability theory
The course is based on Durrett's text book Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
Show more
Related publications (60)

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.