Proposition 6

Let $\nu$ be a measure on U. The restriction of $\nu$ to $\mathscr{D}(U)$ is a distribution, which is zero if and only if the measure $\nu$ is zero.

Let K be a compact subset of U. For every function $\varphi \in \mathscr{C}_K^{\infty}$(U), one has $|\langle \nu , \varphi \rangle |\leqslant p_{0,K}(\varphi )|\nu |$(K), hence the restriction of $\nu$ to $\mathscr{D}(U)$ is continuous. Since $\mathscr{D}(U)$ is dense in $\mathscr{K}(U)$ (prop. 4, a) of IV, p. 202), the restriction of $\nu$ to $\mathscr{D}(U)$ is zero if and only if $\nu$ is zero.

We shall identify the space $\mathscr{M}(U;\mathbf{C})$ of complex measures on U with a subspace of $\mathscr{D}'(U)$. We shall also identify the space $L^1_{loc}(U)$ with a subspace of $\mathscr{D}'(U)$ by the mapping deduced by passing to quotients from the mapping which associates to $f\in \mathscr{L}_{loc}^1(U)$ the measure $f\cdot \mu$ (INT, V, p. 44, § 5, No.$^o2$, def. 2). In other words, for $f\in L^1_{loc}(U)$ and $\varphi \in \mathscr{D}$(U), one has

In particular, for $p\in [1,+\infty ]$, this permits one to identify the space $L^p(U)$ with a subspace of $\mathscr{D}'(U) ($cf. INT, V, p. 43, § 5, No.$^o1)$.