HAAR MEASURE Meaning and
Definition
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Haar measure is a term used in mathematics to refer to a measure on the space of functions, particularly on topological vector spaces or locally compact groups. It is closely related to the concept of Lebesgue measure in Euclidean spaces, but it extends to more general settings.
In simple terms, haar measure is a way to assign a measure to functions that satisfies certain properties. It is often used in the study of Fourier analysis and harmonic analysis, where it provides a way to measure the "size" or "weight" of functions.
Formally, haar measure is a measure that is invariant under translations (or left translations) by elements of the group or vector space. This means that if we translate a function by a specific amount, the haar measure of the translated function remains the same as that of the original function.
Additionally, another important property of haar measure is that it is both outer regular and inner regular. This means that it can be approximated well by open sets and compact sets, respectively.
The term "haar measure" originates from the German mathematician Alfréd Haar, who introduced this concept in 1933. Haar measures have found applications in various areas of mathematics, including representation theory, functional analysis, and number theory, where they play a significant role in understanding the structure and behavior of functions or groups.
Etymology of HAAR MEASURE
The word "haar measure" comes from the Scottish dialect, specifically from the Old Scots word "ha(a)r", which means "hoar" or "hoary" and refers to thick fog or mist. In Scotland, the haar often brings a damp, sea fog that can blanket the coastline and extend inland. The term "haar measure" is used to describe a unit of distance used by fishermen and sailors living along the Scottish coast. It represents the distance that can be covered during the time it takes for a haar to clear or for visibility to improve.
