I'm reading the book "Linear Partial Differential Operators I" by Hormander. I don't understand the identification of a measure as a linear form. The definition of a linear form that appears in the book is the following:
Definition. A distribution $u$ on $X$ is a linear form on $\mathcal{C}_{0}^{\infty}(X)$ such that for every compact set $K\subset X$, there exist constants $C$ and $k$ such that
$$(2.1.2)\qquad\qquad|u(\phi)|\leq C\sum_{|\alpha|\leq k}\sup|\partial^{\alpha}\phi|,\quad \phi\in\mathcal{C}_{0}^{\infty}(K)$$
Subsequently, it is mentioned that:
A measure can be defined as a linear form on $\mathcal{C}_{0}^{0}(X)$ with the continuity property $(2.1.2)$ for $k=0.$
Question. Does the identification of a measure $\mu$ arise from $\mu(\phi)=\int \phi(x)d\mu(x)$? I don't understand that part. Thank you.