This measure was introduced by Alfréd Haar[?], a Hungarian mathematician about 1932.
If G is a locally compact topological group, we can consider the σalgebra X generated by all compact subsets of G. If a is an element of G and S is a set in X, then the set aS = {as : s in S} (where the multiplication is the group operation in G) is also in X. A measure μ on X is called lefttranslationinvariant if μ(aS) = μ(S) for all a and S.
It turns out that there is, up to a multiplicative constant, only one lefttranslationinvariant measure on X which is finite on all compact sets. This is the Haar measure on G. (There is also an essentially unique righttranslationinvariant measure on X, but the two measures need not coincide.) Using the general Lebesgue integration approach, one can then define an integral for all measurable functions f : G > R (or C). This is the beginning of harmonic analysis.
The Haar measure on the topological group (R, +) which takes the value 1 on the interval [0,1] is equal to the Borel measure. This can be generalized for (R^{n}, +). If G is the group of positive real numbers with multiplication as operation, then the Haar measure μ(S) is given by ∫_{S} 1/x dx for any Borel set S.
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