Suppose $K \subset \mathbb{R}^n$ be compact and $\{s_j\}_{j=1}^\infty$ be a countable dense set of $K$. Define, $u_s(x) = \max\{2-\frac{|x-s|}{dist(x,K)},0\}$, $\sigma(x) = \sum_{j=1}^\infty 2^{-j} u_{s_j}(x)$, $v_k(x) = \frac{2^{-k}u_{s_k}(x)}{\sigma(x)}$. All these functions are defined for $x \in K^c$.

The authors (Evans & Gariepy) Page 14 of this book say that $v_k(x)$ form partitions of unity on $K^c$. Using the definition of Partition of Unity, I am unable to show that this is the case, i.e., there exists a neighborhood for every $x\in K^c$, where only finitely many functions $v_k$ are non-zero.

For example, let $K$ be the closed unit ball and choose $x$ far enough from $K$ so that there exits no $s$ for which $|x-s| \geq 2dist(x,K)$ and hence infinitely many functions $v_{s_k}$ (all of them) are non zero at this point $x$.

This result is required to prove Lusin's theorem later on in the book.



Yes, a partition of unity is usually meant to be locally finite, but there is no law against using the term in other ways. The authors never defined partition of unity in the book. It seems that they use it for any collection of nonnegative functions that add up to $1$.

The authors were aware of the fact that the sum is not locally finite, because they refer to the Weierstrass M-test to argue that $\bar f$ is continuous. For a locally finite partition of unity, that would not be needed.


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