Characteristic functions and averages - Azevedo, Assis (2013)
Let $\Omega$ be a set and $\Omega_1,\ldots,\Omega_{m-1}$ subsets of $\Omega$, being $m$ an integer greater than one. For a given function $f=(f_1,\ldots f_m):\Omega\rightarrow\mathbb{R}^m$, we prove the existence of a unique function $\alpha=(\alpha_1,\ldots,\alpha_m):\Omega\rightarrow\mathbb{R}^m$ such that \begin{eqnarray*} \left\{ \begin{array}{lcl} \alpha_i & = & \alpha_{i+1} \quad\mbox{on $\Omega_i$}\\ \alpha_1+\cdots+\alpha_i & = & f_1+\cdots+f_i\quad\mbox{on $\Omega\setminus\Omega_i$, for all $i< m$}\\ \alpha_1+\cdots+\alpha_m & = & f_1+\cdots+f_m, \end{array} \right. \end{eqnarray*} called the average function of $f:\Omega\rightarrow\mathbb{R}^m$ relatively to $\left(\Omega,\Omega_1,\ldots,\Omega_{m-1}\right)$. When $\Omega$ is a topological space and $f$ is a continuous function, we find necessary and sufficient conditions for the continuity of the average function of $f$. We write $\alpha_i$ as a linear combination of characteristic functions of the (coincidence) sets $\cap_{j=r}^s\Omega_j$, $1\leq r\leq s\leq m-1$, belonging the coefficients to $\mathbb{Q}[f_1,\ldots,f_m]$.