Non negative Measurable function and Simple function

[Lily@pie]

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...

 

[mighty2000]

Thank you for your post. To answer your question, let's first define H_{+}(Ω,A) and S_{+}(Ω,A). H_{+}(Ω,A) is the set of all functions f:Ω→[0,∞) that are A/Borel(ℝ) measurable, meaning that their pre-images of Borel sets are measurable. S_{+}(Ω,A) is a subset of H_{+}(Ω,A) consisting of functions with a finite number of values in [0,∞). In other words, the range of these functions is finite and contained in [0,∞).

Now, let's apply these definitions to the given problem. Since Ω=ℝ, any f:Ω→[0,∞) is a function from ℝ to [0,∞). In order for f to be A/Borel(ℝ) measurable, its pre-image of any Borel set in ℝ must be measurable. This means that f must be continuous or discontinuous at a countable number of points. Therefore, for any function f:Ω→[0,∞) in H_{+}(Ω,A), its inverse will either be countable or its complement will be countable.

Next, let's consider S_{+}(Ω,A). Since the range of functions in S_{+}(Ω,A) is finite, the inverse of any Borel set in [0,∞) will also be finite. This means that the inverse of any Borel set in [0,∞) must be countable. Therefore, the complement of the inverse, which is also a Borel set in [0,∞), must also be countable.

In conclusion, H_{+}(Ω,A) consists of functions that are either continuous or discontinuous at a countable number of points, while S_{+}(Ω,A) consists of functions with a finite range and whose inverse and complement of the inverse are both countable.

I hope this helps clarify the problem for you. Let me know if you have any further questions.