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Thread: Sigma-algebra

  1. #1
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    Sigma-algebra

    Hello,

    I have been given the follwoing subsets of $\displaystyle \mathbb{R}$: $\displaystyle A=\left[0,1 \right]$, $\displaystyle B=\left]\tfrac{1}{2},\infty \right[$ and $\displaystyle C= \mathbb{Z}$.

    I wish to show that the sets $\displaystyle \left\{-1,-2,-3,... \right\}$, $\displaystyle \left\{0\right\}$ and $\displaystyle A=\left[0,\tfrac{1}{2}\right] \cup \left\{1\right\} $ belong to the $\displaystyle \sigma$-algebra $\displaystyle \sigma(\left\{A,B,C \right\})$.

    Would it be enough to, tediously, write out the $\displaystyle \sigma$-algebra and check that the subsets are indeed included or is there an easier method?

    Thanks.
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  2. #2
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    Re: Sigma-algebra

    (you are using $A$ to denote two different sets)

    Is this an example of what you are looking for? (not tedious)

    using $A=[0,1]$

    $\displaystyle B^c\cap A\cap C=\{0\}$
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  3. #3
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    Re: Sigma-algebra

    Yes, I see. That was a typo.

    I wish to show that the sets $\displaystyle \left\{-1,-2,-3,... \right\}$, $\displaystyle \left\{ 0 \right\}$ and $\displaystyle \left[0,\tfrac{1}{2} \right] \cup \left\{1 \right\}$ belong to $\displaystyle \sigma(\left\{A,B,C \right\}) $.

    The $\displaystyle \sigma$-algebra contains sets which are obtained by countable unions, intersections and complements of the sets $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$.
    I did an example with two sets only and thought that it was a bit tedious and therefore I wanted to know if there is a quicker method of constructing a $\displaystyle \sigma$-algebra.

    This is what I think:

    By definition $\displaystyle B^{c} \in \sigma(\left\{A,B,C \right\}) $.
    By definition $\displaystyle A \in \sigma(\left\{A,B,C \right\}) $.
    By definition $\displaystyle C\in \sigma(\left\{A,B,C \right\}) $.

    By the properties of a $\displaystyle \sigma$-algebra we have that $\displaystyle B^{c} \cap A \cap C = \left\{ 0 \right\} \in \sigma(\left\{A,B,C \right\})$.

    Using the same method as above we see that $\displaystyle B^{c} \cap A^{c} \cap C = \left\{ -1,-2,-3,... \right\} \in \sigma(\left\{A,B,C \right\})$. Correct?
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  4. #4
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    Re: Sigma-algebra

    Correct
    Thanks from detalosi
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