Proving the Existence of a Map: f, h, and g

In summary, there are two maps f and h that have certain properties and we need to show that there exists a map g which satisfies a certain condition. The key is to use the given assumptions to deduce the necessary properties of f and h and then use this to show the existence of g. Additionally, it is important to consider the assumption about h being measurable and how it relates to the measurability of g.
  • #1
P3X-018
144
0
I have 2 maps f and h such

[tex] f :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Y}, \mathbb{K}) [/tex]
[tex] h :\, (\mathcal{X}, \mathbb{E}) \rightarrow (\mathcal{Z}, \mathbb{G}) [/tex]

where [itex] \mathbb{K} [/itex] and [itex] \mathbb{G} [/itex] are [itex] \sigma[/itex]-algebras on the spaces Y and Z respectively, and [itex] \mathbb{E} = \sigma(f) [/itex] is the [itex] \sigma[/itex]-algebra generated by the map f.
f is assumed to be surjective and [itex] \mathbb{G} [/itex] is assumed to separate points in Z. That is for any 2 different point a and b in Z we can find a set [itex] A\in \mathbb{G} [/itex] such that [itex] a\in A [/itex] and [itex] b \notin A [/itex].

The problem is that, assuming h is [itex]\mathbb{E}[/itex]-[itex]\mathbb{G} [/itex] measurable, to show that there is a map [itex] g:\, \mathcal{Y} \rightarrow \mathcal{Z} [/itex] such that [itex] h = g\circ f [/itex].

I don't even know how to attack this problem. What is even meant by showing that there IS such a map? A hint could be helpful.
 
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  • #2
Can you go from (Y,K) to (X,E)? If you can, then you know how to go from (X,E) to (Z,G).
 
  • #3
So if I could go from (Y,K) to (X,E) that is if there was a inverse map of f, that would be enough to conclude that THERE IS a map g?
But f would have to be bijective inorder to go from Y -> X.
Can f be injective under the given assumption? Because it's already surjective, so if it were injective we would be done.
 
  • #4
You do not assume injectivity.

Clearly the hypotheses have been given for a reason. So start to see what you can deduce from them. Let's see what you need to do. You need, given a point y in Y to find a way to associate it to a point g(z) in Z. And we need to do it in such a way that gf(x)=h(x). Wow, what do you know, from just writing that down I've essentially solved the question for you...
 
  • #5
I didn't mean that I assumed f to be injective, only that wether it should (or could) be shown that f is injective, and if so would that solve the problem, since we could go from Y to Z by h(f^(-1)(y)).
Hmm this sounds too 'simple'...
So basically the argument is, that since f is surjective then for every y in Y there is a x in X, such that y = f(x), and since z = h(x), z is the element in Z that is mapped into by y = f(x), with g(f(x)) = h(x), that is z is mapped by z = g(y).


EDIT:
Another question is to show that g is K-G measurable.
Now if I could look at the map from Y -> X that is [itex] f^{-1}[/itex], then I could use the theorem that says that g is K-G measurable if and only if [itex] f^{-1}[/itex] is K-E measurable, assuming that h is E-G measurable.
I'm pretty certain of that I need to use this theorem, somehow, to show the measurability of g, since obviously I can't investigate wether [itex] g^{-1}(A) \in \mathbb{K},\,\, \forall A\in\mathbb{G} [/itex].
 
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  • #6
What was the other assumption in the original question that you've not used? The one abotu G separating points. PErhaps that is useful.
 

1. What is the significance of proving the existence of a map f, h, and g?

Proving the existence of a map f, h, and g is important because it helps us understand the relationships between different mathematical structures and can aid in solving complex problems in various fields, such as physics, engineering, and computer science.

2. How can we prove the existence of a map f, h, and g?

There are various methods for proving the existence of a map f, h, and g, but one common approach is to use the concept of continuity. By showing that the functions are continuous and satisfy certain conditions, we can demonstrate that the map does indeed exist.

3. What are the key components to proving the existence of a map f, h, and g?

The key components to proving the existence of a map f, h, and g are demonstrating continuity, defining the domain and range of each function, and showing that the functions satisfy certain properties, such as being one-to-one or onto.

4. Can the existence of a map f, h, and g be proven for any mathematical structure?

Yes, the existence of a map f, h, and g can be proven for any mathematical structure as long as the functions used satisfy the necessary conditions for the map to exist. However, the specific methods for proving the existence may vary depending on the type of structure being studied.

5. What are some real-world applications of proving the existence of a map f, h, and g?

The concept of proving the existence of a map f, h, and g has applications in many fields, such as in cryptography, where it is used to ensure the security of communication systems. It is also used in physics and engineering to model and understand complex systems, and in computer science for optimization and algorithm design.

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