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Thread: sigma-rings

  1. #1
    Senior Member Sampras's Avatar
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    sigma-rings

    Suppose $\displaystyle \mathfrak{R} $ is a $\displaystyle \sigma $-ring. Suppose $\displaystyle E_n \in \mathfrak{R} $. How would we show that $\displaystyle \bigcap_{n=1}^{\infty} E_n \in \mathfrak{R} $?

    We know that $\displaystyle \bigcup_{n=1}^{\infty} E_n = \bigcap_{n=1}^{\infty} E_{n}^{c} \in \mathfrak{R} $. Likewise, how would we show that $\displaystyle \overline{\lim} \ E_n \in \mathfrak{R} $ and $\displaystyle \underline{\lim} \ E_n \in \mathfrak{R} $? The former is the set of points that are in infinitely many $\displaystyle E_n $. The latter is the set of points that are in all but a finite number of $\displaystyle E_n $.
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  2. #2
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    1)For difference, do not use the whole space X, use $\displaystyle \bigcup \limits_{n =1}^\infty E_n$ instead.
    2)Another equivalent definition of $\displaystyle \overline{\lim} \ E_n$ is $\displaystyle \bigcap \limits_{k = 1}^\infty \bigcup \limits_{n = k}^\infty E_n$, and $\displaystyle \underline{\lim} \ E_n$ is $\displaystyle \bigcup \limits_{k = 1}^\infty \bigcap \limits_{n = k}^\infty E_n$,
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