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Math Help - 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:

    $1_{A}\left(\omega\right)=\begin{cases} 1, & \omega\in A\\ 0, & \omega\in A^{c} \end{cases}$
    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
    Last edited by entropyslave; November 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
    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 \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 View Post
    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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