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.