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Non negative Measurable function and Simple function

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Ω=ℝ
    A=σ({x}[itex]:x\in ℝ[/itex]})

    Determine [itex]H_{+}(Ω,A)[/itex] and [itex]S_{+}(Ω,A)[/itex]

    2. Relevant equations

    [itex]H_{+}(Ω,A)[/itex] is the set of f:Ω→[0,∞) such that f is A/Borel(ℝ) measurable

    [itex]S_{+}(Ω,A)[/itex] is the set of function in [itex]H_{+}(Ω,A)[/itex] such that number of f(Ω) is finite and [itex]f(Ω) \subseteq [0,∞)[/itex]

    3. The attempt at a solution
    I try to break down the requirements of the function and knowing that A is a set that consists of sets that is countable or the complement is countable by part and obtain the following

    For all f:Ω→[0,∞) in [itex]H_{+}(Ω,A)[/itex], [itex]f^{-1}(B)[/itex] or [itex](f^{-1}(B))^{C}[/itex] is countable for all B in the Borel field.

    I'm not sure how to proceed from here. What I have in mind is when the function that maps from dots or a constant function discontinued at a few points? Since if the function is map from dots, the inverse will be countable. And the complement of the inverse will be countable if the function is map from a constant function discontinued at a few points.

    But I'm not sure if this is just one of the many kind of function.

    Thanks a lot in advance....
     
  2. jcsd
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