# Indicator function confusion

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• Nov 6th 2011, 07:40 AM
entropyslave
Indicator function confusion
Hi,

I'm currently reading a book on probability theory (Resnick, "A Probability Path") and in a section on basic set and measure theory, the indicator function is defined as follows:

$1_{A}\left(\omega\right)=\begin{cases} 1, & \omega\in A\\ 0, & \omega\in A^{c} \end{cases}$
and some properties are discussed, including:

$1_{\cup_{n}A_{n}}\le\underset{n}{\sum}1_{A_{n}}$
for which it is stated: "and if the sequence $\left\{ A_{n}\right\}$ 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
$1_{\inf_{n\geq k}A_{n}}=\underset{n\geq k}{\inf}1_{A_{n}}$

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 $\omega$, $\omega\in A_{n}$ for all $n\ge k$ which is also true for the RHS. However, I don't understand how this equality holds since if $\omega\in A_{n}$ for some but not all $n$, 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 $U$ so that $A_n\subseteq U$ for all $n$. Then if $F$ is a family of functions from $U$ to $\mathbb{R}$ (or from $U$ to {0, 1} in this case), $\inf F$ is probably defined pointwise: $(\inf F)(\omega)=\inf\{f(\omega)\mid f\in F\}$ for every $\omega\in U$. So, $(\inf_{n\ge k} 1_{A_n})(\omega)=\inf_{n\ge k} (1_{A_n}(\omega))$, and it equals 1 iff $1_{A_n}(\omega)=1$ for all $n\ge k$.

Quote:

Originally Posted by entropyslave
Another relation introduced is:
$1_{\cup_{n}A_{n}}\le\underset{n}{\sum}1_{A_{n}}$
for which it is stated: "and if the sequence $\left\{ A_{n}\right\}$ 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 $\left\{ A_{n}\right\}$ is mutually disjoint. The left-hand side equals 1 when $\omega$ belongs to any of $A_n$, whereas in the right-hand side, you add 1 for each $A_n$ to which $\omega$ belongs.

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