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Thread: Show whether E is a Measurable Set

  1. #1
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    Show whether E is a Measurable Set

    Let $\displaystyle (E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\displaystyle \mathbb{R}^n$. Let $\displaystyle E$ be a subset of $\displaystyle \mathbb{R}^n$ so that $\displaystyle x \in E$ if and only if $\displaystyle x$ belongs to exactly 2 of the sets $\displaystyle E_k$. Determine whether E is measurable. If it is, show it. If not, give a counterexample.

    I suppose it is measurable? No? Can't think of any counterexample.

    Thanks in advance.
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  2. #2
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    Re: Show whether E is a Measurable Set

    for any two positive integers $\displaystyle 1\leq i <j$ consider the set $\displaystyle A_{i,j}=E_i\bigcap E_j\bigcap (\bigcap_{k\ne i,j}\mathbb{R}^n\setminus E_k) $. Such sets are measurable and $\displaystyle E$ is a union of such sets
    Last edited by Georgii; Oct 13th 2011 at 10:13 AM.
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  3. #3
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    Re: Show whether E is a Measurable Set

    Quote Originally Posted by Markeur View Post
    Let $\displaystyle (E_k)_{k=1}^{\infty}$ be a sequence of measurable sets in $\displaystyle \mathbb{R}^n$. Let $\displaystyle E$ be a subset of $\displaystyle \mathbb{R}^n$ so that $\displaystyle x \in E$ if and only if $\displaystyle x$ belongs to exactly 2 of the sets $\displaystyle E_k$. Determine whether E is measurable. If it is, show it. If not, give a counterexample.

    I suppose it is measurable? No? Can't think of any counterexample.

    Thanks in advance.
    In my opinion any question about "something" should start off with a definition of "something" and a demonstration that the definition is satis fied. So here is my little contribution as a matter of principle from http://en.wikipedia.org/wiki/Measure_(mathematics)

    σ-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable.


    Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
    • Non-negativity:
    for all

    • Null empty set:

    I would find it interesting to see how this would be done in the case of the previous post.
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