Definition 5

A measure $\nu$ on $\mathbf{R}^n$ is said to be tempered if there exists an integer $r\in \mathbf{N}$ such that the continuous mapping $x\mapsto (1 +\|x\|)^{-r}$ is $\nu$-integrable on $\mathbf{R}^n$.

In other words, a measure $\nu$ is tempered if there exists $r\in \mathbf{N}$ such that the function defined by $x\mapsto \|x\|^{-r}$ is $\nu$-integrable on the complement of the unit ball in $\mathbf{R}^n$. In particular, every bounded measure on $\mathbf{R}^n$ is tempered. More generally, if $f$ is a $\mu$-measurable function of polynomial growth and if $\nu$ is tempered, then the measure $f\cdot \nu$ is tempered.

The set $\mathscr{M}^t(\mathbf{R}^n)$ of tempered measures on $\mathbf{R}^n$ is a subspace of the vector space $\mathscr{M}(\mathbf{R}^n;\mathbf{C})$ of complex measures on $\mathbf{R}^n$.