Results 1 to 2 of 2

Math Help - Sigma Algebra is closed under countable increasing unions

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Sigma Algebra is closed under countable increasing unions

    Prove that an algebra  \mathbb {A} is a  \sigma -algebra iff  \mathbb {A} is closed under countable increasing unions.

    Proof so far:

    Suppose that  \mathbb {A} is a  \sigma -algebra, then if E_1,E_2,... \subset \mathbb {A} , then I have  \bigcup _{n=1}^{ \infty } E_n \subset \mathbb {A}

    Now, if E_1 \subset E_2 \subset E_3 ... \subset \mathbb {A} , then  \bigcup _{n=1}^{ \infty } E_n \subset \mathbb {A} since E_n \in \mathbb {A} \ \ \ \forall n. This direction is simple.

    On the other hand, if A is closed under countable increasing unions, that means that for:

    E_1 \subset E_2 \subset E_3 \subset ... \subset \mathbb {A} , then I have  \bigcup _{n=1}^{ \infty } E_n \subset \mathbb {A} And I need to show that  \mathbb {A} is a  \sigma -algebra.

    Suppose that E_1, E_2, ... \subset \mathbb {A} , I need to show that:

    1. E^c \in \mathbb {A}

    2. \bigcup _{n=1}^{ \infty } E_n \subset \mathbb {A}

    How should I proceed from here? Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2008
    Posts
    81
    Well being closed under complements is a property of an algebra which \mathbb{A} is, thus the first part is a moot point.
    As to countable unions, consider F_n=\displaystyle\bigcup_{j=1}^n E_j which is a countable increasing sequence of sets.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: February 4th 2011, 09:39 AM
  2. 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
  3. Sigma algebra, open and closed set.
    Posted in the Differential Geometry Forum
    Replies: 12
    Last Post: November 10th 2010, 09:26 AM
  4. [SOLVED] Countable union of closed sets/countable interesection of open sets
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 8th 2010, 02:59 PM
  5. Infintie unions of closed sets in R
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 22nd 2008, 04:20 PM

Search Tags


/mathhelpforum @mathhelpforum