Proposition 15

Let $\nu$ be a tempered measure 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.

Since $\nu$ is tempered, there exists a positive integer $k$ such that the mapping $x\mapsto \|x\|^{-k}$ is $\nu$-integrable on the complement of the unit ball in $\mathbf{R}^n$. For every Schwartz function $\varphi \in \mathscr{S}(\mathbf{R}^n)$, one has

therefore the restriction of $\nu$ to $\mathscr{S}(\mathbf{R}^n)$ is a tempered distribution.

The last assertion follows from Prop. 6 of IV, p. 206 since $\mathscr{D}(\mathbf{R}^n)$ is contained in $\mathscr{S}(\mathbf{R}^n)$.

We shall identify the space $\mathscr{M}^t(\mathbf{R}^n)$ with a subspace of $\mathscr{S}'(\mathbf{R}^n)$.