10. Integration of differential forms

Partitions of unity

Theorem 10.8

• $K \subseteq \mathbb{R}^{n}$,
  • compact
• \(\{V_{\alpha}\}\),
  • open cover of $K$,

Then there exist functions $\psi_{1}, \ldots, \psi_{s} \in \mathsrc{C}(\mathbb{R}^{n}$ such that

• (1) $0 \le \psi_{i} \le 1$, for all $i$,
• (2) $\exists \alpha_{i}$ such that $\mathrm{supp}\phi_{i} = V_{\alpha_{i}}$,
• (3) $\psi_{1}(x) + \cdots + \psi_{s}(x) = 1$ for every $x \in K$,

proof

For all $x \in K$, there exists an index $\alpha(x)$ such that $x \in V_{\alpha(x)}$.

\[\begin{equation} \overline{B(x)} \subseteq W(x) \subseteq \overline{W(x)} \subseteq V_{\alpha(x)} \label{equation_09_26} \end{equation} .\]

Since $K$ is compat, there are points $x_{1}, \ldots, x_{s} \in K$ such that

\[K \subseteq B(x_{1}) \cup \cdots \cup B(x_{s}) .\]

By \(\eqref{equation_09_26}\), there are functions $\phi_{1}, \ldots, \phi_{s} \in \mathscr{C}(\mathbb{R}^{n}$ such that

\[\phi_{i}(x) = 1 .\]

$\Box$

Corollary

• $f \in \mathscr{C}(\mathbb{R}^{n})$,
  • $\mathrm{supp}f \subseteq K$,

\[f = \sum_{i=1}^{s} \psi_{i}f .\]

proof

$\Box$