Results 1 to 2 of 2

Math Help - sigma-rings

  1. #1
    Senior Member Sampras's Avatar
    Joined
    May 2009
    Posts
    301

    sigma-rings

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

    We know that  \bigcup_{n=1}^{\infty} E_n = \bigcap_{n=1}^{\infty} E_{n}^{c} \in \mathfrak{R} . Likewise, how would we show that  \overline{\lim} \ E_n \in \mathfrak{R} and  \underline{\lim} \ E_n \in \mathfrak{R} ? The former is the set of points that are in infinitely many  E_n . The latter is the set of points that are in all but a finite number of  E_n .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2009
    Posts
    120
    1)For difference, do not use the whole space X, use \bigcup \limits_{n =1}^\infty E_n instead.
    2)Another equivalent definition of \overline{\lim} \ E_n is \bigcap \limits_{k = 1}^\infty \bigcup \limits_{n = k}^\infty E_n, and \underline{\lim} \ E_n is \bigcup \limits_{k = 1}^\infty \bigcap \limits_{n = k}^\infty E_n,
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring theory, graded rings and noetherian rings
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 4th 2012, 12:46 PM
  2. a book on semigroup rings and group rings
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 2nd 2011, 05:35 AM
  3. Replies: 1
    Last Post: February 4th 2011, 09:39 AM
  4. Show intersection of sigma-algebras is again a sigma-algebra
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 20th 2010, 08:21 AM
  5. how to determine sigma and sgn(sigma) for matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 16th 2010, 12:12 PM

Search Tags


/mathhelpforum @mathhelpforum