Definition For Locally Compact Groups
Let be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of second countable) measure (left and right probability measure in the case of compact), the Haar measure. Consider the banach space of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
Definition 1. A linear functional is said to be a mean if has norm 1 and is non-negative (i.e. a.e. implies ).
Definition 2. A mean is said to be left-invariant (resp. right-invariant) if all with respect to the left (resp. right) shift action of (resp. ).
Definition 3. A locally compact hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
Read more about this topic: Amenable Group
Famous quotes containing the words definition, locally, compact and/or groups:
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“The powers of the federal government ... result from the compact to which the states are parties, [and are] limited by the plain sense and intention of the instrument constituting that compact.”
—James Madison (1751–1836)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)