Example

A measure $\nu$ on $\mathbf{R}^n$ is said to be tempered if there exists a positive integer $r$ such that the continuous mapping $x\mapsto (1+\|x\|)^{-r}$ is $\nu$-integrable on $\mathbf{R}^n$. The restriction of $\nu$ to $\mathscr{S}(\mathbf{R}^n)$ is a tempered distribution. It is zero if and only if the measure $\nu$ is zero.

Let $p\in [1,+\infty ]$ and $f\in \mathscr{L}^p(\mathbf{R}^n)$. Then the measure $f\cdot dx$ with density $f$ with respect to Lebesgue measure is tempered. In particular, Lebesgue measure $\mu$ on $\mathbf{R}^n$ is tempered, and every bounded measure on $\mathbf{R}^n$ is tempered.

For every $p\in [1,+\infty ]$, one can identify $L^p(\mathbf{R}^n)$ with a subspace of $\mathscr{S}'(\mathbf{R}^n)$ by the linear mapping $f\mapsto f\cdot dx$; this mapping is continuous.