> [!theorem] > > Let $X$ be a [[Locally Compact Hausdorff Space|LCH]] space and $\seqi{U}$ be an [[Open Cover|open cover]] of $X$. Suppose that > 1. For each $i \in I$, there exists a [[Linear Functional|functional]] $\phi_i \in C_c(U_i)^*$ on the [[Compactly Supported|compactly supported]] continuous functions. > 2. Whenever $U_i \cap U_j \ne \emptyset$, $\phi_i|_{C_c(U_i \cap U_j)} = \phi_j|_{C_c(U_i \cap U_j)}$. > > Then there exists a linear functional $\phi \in C_c(X)^*$ such that $\phi|_{C_c(U_i)} = \phi_i$ for all $i \in I$. If each $\phi_i$ is [[Positive Linear Functional|positive]], then $\phi$ is positive as well. > > *Proof*. Let $f \in C_c(X)$ and $K = \supp{f}$. Let $\seqf{U_j}$ be a finite subcover of $K$, $\seqf{\phi_j}$ be the associated functionals, and $\seqf{\chi_j}$ be a [[Partition of Unity|partition of unity]] subordinate to it. Define > $ > \angles{\phi, f} = \sum_{j = 1}^n \angles{\phi_j, \chi_jf} > $ > Let $\bracs{V_j}_1^m$ be another finite subcover of $K$, $\bracs{\psi_j}_1^m$ be the associated functionals, and $\bracs{\omega_j}_1^n$ be a partition of unity subordinate to it. In this case, for each fixed $1 \le i \le n$, by assumption $(2)$, > $ > \angles{\phi_i, \chi_if} = \sum_{j = 1}^m\angles{\phi_i, \chi_i\omega_jf} = \sum_{j = 1}^m \angles{\psi_j, \chi_i\omega_jf} > $ > On the flip side, for each fixed $1 \le j \le m$, > $ > \angles{\psi_j, \omega_j f} = \sum_{i = 1}^n\angles{\psi_j, \chi_i\omega_jf} = \sum_{i = 1}^n\angles{\phi_i, \chi_i\omega_jf} > $ > Therefore > $ > \begin{align*} > \sum_{i = 1}^n\angles{\phi_i, \chi_if} &= \sum_{i = 1}^n\sum_{j = 1}^m\angles{\phi_i, \chi_i\omega_jf} \\ > &= \sum_{i = 1}^n\sum_{j = 1}^m\angles{\psi_j, \chi_i\omega_jf} = \sum_{j = 1}^m\angles{\psi_j, \omega_j f} > \end{align*} > $ > and this is well-defined. The restriction property comes from the well-defined property.