partition of unity

Let $X$ be a topological space. A partition of unity is a collection of continuous functions $\{{\epsilon }_i:X\to [0,1]\}$ such that

$$\sum _i{\epsilon }_i(x)=1\mathit{\hspace{1em}}\text{for all }x\in X.$$ (1)

A partition of unity is locally finite if each $x$ in $X$ is contained in an open set on which only a finite number of ${\epsilon }_i$ are non-zero. That is, if the cover $\{{\epsilon }_i^{-1}((0,1])\}$ is locally finite.

A partition of unity is subordinate to an open cover $\{U_i\}$ of $X$ if each ${\epsilon }_i$ is zero on the complement of $U_i$.

Example 1 (Circle)

A partition of unity for ${\mathrm{S}}^{\mathrm{1}}$ is given by $\mathrm{\{}{\sin }^{\mathrm{2}}\mathit{}\mathrm{(}\theta \mathrm{/}\mathrm{2}\mathrm{)}\mathrm{,}{\cos }^{\mathrm{2}}\mathit{}\mathrm{(}\theta \mathrm{/}\mathrm{2}\mathrm{)}\mathrm{\}}$ subordinate to the covering $\mathrm{\{}\mathrm{(}\mathrm{0}\mathrm{,}\mathrm{2}\mathit{}\pi \mathrm{)}\mathrm{,}\mathrm{(}\mathrm{-}\pi \mathrm{,}\pi \mathrm{)}\mathrm{\}}$.

Application to integration

Let $M$ be an orientable manifold with volume form $\omega$ and a partition of unity $\{{\epsilon }_i(x)\}$. Then, the integral of a function $f(x)$ over $M$ is given by

$${\int }_{M}f(x)\omega =\sum _i{\int }_{U_i}{\epsilon }_i(x)f(x)\omega .$$

It is of the choice of partition of unity.

Title	partition of unity
Canonical name	PartitionOfUnity
Date of creation	2013-03-22 13:29:23
Last modified on	2013-03-22 13:29:23
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	10
Author	mhale (572)
Entry type	Definition
Classification	msc 54D20
Classification	msc 58A05
Defines	locally finite partition of unity
Defines	subordinate to an open cover