This measure was introduced by Alfréd Haar, a Hungarian mathematician about 1932.

If *G* is a locally compact topological group, we can consider the σ-algebra *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 *left-translation-invariant* if μ(*aS*) = μ(*S*) for all *a* and *S*.

It turns out that there is, up to a multiplicative constant, only one left-translation-invariant measure on *X* which is finite on all compact sets. This is the Haar measure on *G*. (There is also an essentially unique right-translation-invariant 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**), called the **Haar integral**. 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* d*x* for any Borel set *S*.