Counting measure as Haar measure on a discrete group

Every subset of $G$ is open and Borel. A compact subset is finite, since its cover by singleton open sets has a finite subcover; a finite set is compact by selecting one member of a cover for each point. Disjoint countable additivity of counting follows as follows: if the union is finite, cardinalities add finitely; if it is infinite, either a member is infinite or the finite partial sums of the cardinalities are unbounded, and both sides are infinity.

Outer regularity holds by taking the open superset $E$ itself. The supremum of the sizes of finite subsets of $E$ is $\mathrm{\mid }E\mathrm{\mid }$ if finite and infinity otherwise, since an infinite set has $n$ distinct elements for each finite $n$. This proves open inner regularity. Compact finiteness follows from step 1.1 and $\mu (\{e\})=1$ proves nonzeroness. The bijections $x\mapsto ax$ and $x\mapsto xa$ preserve finite cardinality and infinitude, proving both invariances.

A compactly supported function has finite support, so simple-function integration gives the displayed finite sum. For instance $f={\mathbf{21}}_{\{e\}}$ has $\int fd\mu =2$ and $\int L_{a}fd\mu =\int {\mathbf{21}}_{\{a\}}d\mu =2$ for every $a$. For the zero function the empty sum is zero.