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Non negative Measurable function and Simple function

  • Thread starter Lily@pie
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Homework Statement


Ω=ℝ
A=σ({x}[itex]:x\in ℝ[/itex]})

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

Homework 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]

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....
 

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