Structure theorem for tempered distributions

Let $\mu$ be a measure on $\mathcal{B}\left(\mathbb {R}^{m}\right)$, which attains finite values on compact sets. Prove that the equality $\alpha_{\mu }(\varphi ) = \int_{\mathbb {R}^{m}} \varphi (t) d \mu (t), \varphi \in \mathcal{D}\left(\mathbb {R}^{m}\right)$, defines a generalized function $\alpha_{\mu } \in \mathcal{D}^{\prime }\left(\mathbb {R}^{m}\right)$.

Let $\mu$ be a measure on $\mathcal{B}\left(\mathbb {R}^{m}\right)$, for which there exists $k \in \mathbb {N}$ such that $(1+ \left\| x\right\| )^{-k} \in L_{1}\left(\mathbb {R}^{m}, \mu \right)$ (such a measure is called a tempered measure). Prove that the formula from item (1) defines a tempered distribution $\alpha_{\mu } \in S^{\prime }\left(\mathbb {R}^{m}\right)$.

Show that not every measure, which attains finite values on compacts, defines $\alpha_{\mu } \in \mathcal{S}^{\prime }\left(\mathbb {R}^{m}\right)$.