# Non negative Measurable function and Simple function

## Homework Statement

Ω=ℝ
A=σ({x}$:x\in ℝ$})

Determine $H_{+}(Ω,A)$ and $S_{+}(Ω,A)$

## Homework Equations

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

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

## 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 $H_{+}(Ω,A)$, $f^{-1}(B)$ or $(f^{-1}(B))^{C}$ 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....

## Answers and Replies

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