Compositions of measurable mappings

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In summary, to prove that compositions of measurable mappings are measurable, you need to show that for any measurable set E in the range of the composition, the preimage of E under the composition is a measurable set. This follows from the fact that both mappings in the composition are measurable, and the preimage of a measurable set under a measurable mapping is also measurable.
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Homework Statement


I have to prove that compositions of measurable mappings are measurable.

i.e. If X is [tex]F/\widetilde{F}[/tex] measurable and Y is [tex]\widetilde{F}/\widehat{F}[/tex] measurable, then Z:=YoX:[tex]\Omega\rightarrow\widehat{\Omega}[/tex] is [tex]F/\widehat{F}[/tex] measurable.

Homework Equations


X is [tex]F/\widetilde{F}[/tex] measurable if [tex]X^{-1}(\widetilde{F})=(\omega \in \Omega: X(\omega)\in\widetilde{F})\in F}[/tex]
(that last F is meant to be a curly F, sigma algebra, and the brackets before the little omega and before the last "element of" are meant to be braces.)

The Attempt at a Solution


I know you're not supposed to help if I haven't attempted it, but I've never been great at proofs and honestly don't know where to start. Can anyone give me a nudge in the right direction?
 
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  • #2
Okay, so I came up with something that seems wrong, but can someone tell me if it holds?

[tex]Z^{-1}(\hat{F})=X^{-1}(Y^{-1}(\hat{F}))[/tex]
[tex]=X^{-1}(({\tilde{\omega}\in\tilde{\Omega}:Y(\tilde{\omega})\in\hat{F}))[/tex]
[tex]=(\omega\in\Omega:X(\omega)\in\tilde{F})\in F[/tex]
[tex]=(\omega\in\Omega:Z(\omega)\in\hat{F})\inF[/tex] (since Z:YoX)

Again, some of those brackets are meant to be braces...
 
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  • #3
I'm going to relabel the domains as X, Y, Z and the functions f:X -> Y, g:Y -> Z.

You need to show

(f o g):X -> Z

has the property that for every measurable set E in Z, the preimage (f o g)^{-1}(E) is a measurable set in X.

You know this property holds for both f and g (as they are measurable). What remains is show it holds for the composition.
 

1. What are compositions of measurable mappings?

Compositions of measurable mappings refer to the combination or chaining of multiple measurable mappings to create a new measurable mapping. This allows for the evaluation and measurement of more complex functions or transformations.

2. Why are compositions of measurable mappings important?

Compositions of measurable mappings are important because they allow for the analysis and understanding of more complex mathematical functions and transformations. They also provide a framework for solving real-world problems and making predictions based on data.

3. How are compositions of measurable mappings different from compositions of general functions?

Compositions of measurable mappings differ from compositions of general functions in that measurable mappings are specifically defined for measurable spaces, which have certain properties and structures that allow for the measurement of sets and functions.

4. What is the role of measurable spaces in compositions of measurable mappings?

Measurable spaces play a crucial role in compositions of measurable mappings as they provide the underlying structure for the measurable mappings to operate on. They also allow for the measurement and comparison of sets and functions in a rigorous and well-defined manner.

5. How are compositions of measurable mappings used in scientific research?

Compositions of measurable mappings are used in scientific research to analyze and understand complex systems or processes. They are particularly useful in fields such as physics, engineering, and economics where there is a need to model and predict the behavior of systems using mathematical functions and transformations.

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