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Math Help - criterion for borel sets

  1. #1
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    criterion for borel sets

    i am trying to show that for a function f:X->R to be borel-measurable, it is sufficient that one of the following conditions is met:

    1. f^(-1)(]a,b[) is in A for all a,b in R, a<b where A is the set of measurable sets.
    2. f^(-1)([a,b]) is in A
    3. f^(-1)(]a,+infinite[) is in A
    4. f^(-1)(]-infinite,a[) is in A
    5. f^(-1)([a,+infinite[) is in A
    6. f^(-1)(]-infinite,a]) is in A

    i know that if i can show 1 then it shouldnt be hard to show 1 implies 2, 2 implies 3, etc. since i already showed that each set can be generated by complements, unions, and countable intersections of other sets, but i can't seem to show that condition 1 is true. some help please? thanks.
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  2. #2
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    For condition 1, use the fact that every open subset of R is a countable union of open intervals.
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