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Math Help - Borel algebra over the real line

  1. #1
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    Borel algebra over the real line

    Show that the Borel algebra over  \mathbb {R} , denoted  \mathbb {B} _ \mathbb {R} is generated by the set of  \xi= \{ [a, \infty ) : a \in \mathbb {R} \}

    Set up:

    I know that  \mathbb {B} _ \mathbb {R} = \mathbb {M} (U), where  \mathbb {M} (U) is the smallest  \sigma -Algebra generated by U, and U is the family of all open sets in  \mathbb {R} . So by a theorem that I know, if I can prove that  E \in \mathbb {M} (F) , then I have  \mathbb {M} (E) \subset \mathbb {M} (F) . My plan is to prove that  \xi \in \mathbb {M}(U) and vice versa.

    Proof.

    Claim:  \mathbb {M} ( \xi ) \subseteq \mathbb {M} (U)

    Now, pick an element  [a, \infty ) \in \xi ,
    and write  [a, \infty ) = \bigcup _{r>0} [a,a+r) = \bigcup _{r>0} \bigcap _{n=1}^{ \infty } (a- \frac {1}{n} , a+r )

    Note that  (a- \frac {1}{n} ,a+r) \in U \subset \mathbb {M}(U) \ \ \ \ \ \forall n \in \mathbb {N}, so  \bigcap _{n=1}^{ \infty } (a- \frac {1}{n} , a+r ) \in \mathbb {M}(U).

    Therefore  \bigcup _{r>0} \bigcap _{n=1}^{ \infty } (a- \frac {1}{n} , a+r ) \in \mathbb {M}(U)

    So I have  \xi \in \mathbb {M}(U), which implies that  \mathbb {M} ( \xi ) \subset \mathbb {M} (U)

    On the other hand, pick (a,b) \in U , then  (a,b) = \bigcap _{a<r<b} \bigcup _{n=1}^{ \infty } [r+ \frac {1}{n} , \infty )  \in \mathbb {M} ( \xi ) for the same reason as above.

    Therefore I have  \mathbb {M}(U) \subset \mathbb {M}( \xi ) .

    Is this correct? Thank you.
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  2. #2
    Moo
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    Hello,

    For the first part, it would have been shorter to say that [a,\infty) is the complement of (-\infty,a), which can be written as \bigcup_{n> 0} (a+n,a), which belongs to \mathbb{M}(U)
    But what you did is very correct too !

    On the other hand, pick (a,b) \in U, then (a,b) = \bigcap _{a<r<b} \bigcup _{n=1}^{ \infty } [r+ \frac {1}{n} , \infty ) \in \mathbb {M} ( \xi ) for the same reason as above.
    Hmm no.
    Because the intersection is equal to (b,\infty) or something like that.

    Let :
    A_n=\left[a-\frac 1n,\infty\right) and B=[b,\infty)

    Consider C=\left(\bigcup_{n>0} A_n\right)\cap B^c


    Can you finish it ?


    Anyway, good work you've done !
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