Differentiable Partitions of Unity

This is an extremely useful technique borrowed from Differential Geometry (or is it the other way around?). The idea is to construct a family of compactly supported smooth functions which “add up to one” in a particular set. There are several alternative ways to define “Partitions of Unity”, depending mostly on the nature of the objects in which we work (e.g., differential manifolds vs. compact set in measurable spaces). I will present here one of the most basic results:

Theorem Let $X \subset \mathbb{R}^d$ be an open set, and let $K \subset X$ be a compact subset. Let $\{ X_k \}_{k=1}^m$ be open subsets of $X$ whose union contains $K$ . Then one can find functions $\phi_1, \dotsc, \phi_m$ such that

$\phi_k \in C_c^\infty (X_k)$ , and $0 \leq \phi_k \leq 1$ for all $k=1, \dotsc, m$ .

$\sum_{k=1}^m \phi_k \leq 1$ on $X$ , and $\sum_{k=1}^m \phi_k = 1$ on a neighborhood of $K$ .