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Thread: Indicator function confusion

  1. #1
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    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:

    $\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
    Last edited by entropyslave; Nov 6th 2011 at 08:12 AM. Reason: Change in understanding
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  2. #2
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    Re: Indicator function confusion

    Quote Originally Posted by entropyslave View Post
    $\displaystyle 1_{\inf_{n\geq k}A_{n}}=\underset{n\geq k}{\inf}1_{A_{n}}$
    What is the definition of the infimum of a family of sets? Does it mean intersection?

    Quote Originally Posted by entropyslave View Post
    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 View Post
    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.

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