# Indicator function confusion

$\displaystyle$1_{A}\left(\omega\right)=\begin{cases} 1, & \omega\in A\\ 0, & \omega\in A^{c} \end{cases}$$and some properties are discussed, including: \displaystyle 1_{\cup_{n}A_{n}}\le\underset{n}{\sum}1_{A_{n}}$$
for which it is stated: "and if the sequence $\displaystyle$\left\{ A_{n}\right\}$is mutually disjoint, then equality holds." However, I don't understand this: the LHS can only ever be 0 or 1, whereas the RHS can be any integer between 0 and the total number of subsets. How could equality ever hold except under specific cases? Update: Actually, I just realised I'm being stupid. If the sets are disjoint, then the RHS would also only return 0 or 1. Any pointers greatly appreciated! Chris • Nov 6th 2011, 08:19 AM emakarov Re: Indicator function confusion Quote: Originally Posted by entropyslave$\displaystyle 1_{\inf_{n\geq k}A_{n}}=\underset{n\geq k}{\inf}1_{A_{n}}$What is the definition of the infimum of a family of sets? Does it mean intersection? Quote: Originally Posted by entropyslave There is a proof given for this property that is based on proving the LHS is 1 iff for some element$\displaystyle \omega$,$\displaystyle \omega\in A_{n}$for all$\displaystyle n\ge k$which is also true for the RHS. However, I don't understand how this equality holds since if$\displaystyle \omega\in A_{n}$for some but not all$\displaystyle n$, then the RHS will be the null set (whereas the LHS is only 0 or 1). The RHS is a number, not a set. I assume that there is a universal set$\displaystyle U$so that$\displaystyle A_n\subseteq U$for all$\displaystyle n$. Then if$\displaystyle F$is a family of functions from$\displaystyle U$to$\displaystyle \mathbb{R}$(or from$\displaystyle U$to {0, 1} in this case),$\displaystyle \inf F$is probably defined pointwise:$\displaystyle (\inf F)(\omega)=\inf\{f(\omega)\mid f\in F\}$for every$\displaystyle \omega\in U$. So,$\displaystyle (\inf_{n\ge k} 1_{A_n})(\omega)=\inf_{n\ge k} (1_{A_n}(\omega))$, and it equals 1 iff$\displaystyle 1_{A_n}(\omega)=1$for all$\displaystyle n\ge k$. Quote: Originally Posted by entropyslave Another relation introduced is:$\displaystyle 1_{\cup_{n}A_{n}}\le\underset{n}{\sum}1_{A_{n}}$for which it is stated: "and if the sequence$\displaystyle \left\{ A_{n}\right\}$is mutually disjoint, then equality holds." However, I don't understand this: the LHS can only ever be 0 or 1, whereas the RHS can be any integer between 0 and the total number of subsets. How could equality ever hold except under specific cases It does hold under specific cases: when the sequence$\displaystyle \left\{ A_{n}\right\}$is mutually disjoint. The left-hand side equals 1 when$\displaystyle \omega$belongs to any of$\displaystyle A_n$, whereas in the right-hand side, you add 1 for each$\displaystyle A_n$to which$\displaystyle \omega$belongs. When using the forum's [TEX]...[/TEX] tags, it is not necessary to use$.