Question: Let ( X , ) be a measurable space. Show that f : X R is -measurable if and only if r Q : f

Let(X,) be a measurable space. Show that f:XR is -measurable if and only if rQ:f1((r,)). REFS

Definition of \Sigma-measurable Let be a -measurable on a nonempty set X. Then(X,) is a measurable space. The setAX is -measurable ifA.

Measurable Functions https://people.math.gatech.edu/~heil/6337/spring11/section3.1.pdf https://sites.ualberta.ca/~rjia/Math417/Notes/chap5.pdf

Definitions https://www.bauer.uh.edu/rsusmel/phd/sR-0.pdf

Borel Measurable Functions Let(X,p) be a metric space and B(X)be the Borel -algebra on X. A B(X)-measurable functionf:XR is called Borel measurable s.t AB(R):f1(A)B(X).

Sigma Algebra https://mathworld.wolfram.com/Sigma-Algebra.html

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