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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?