Let be some measurable domain and f : E → R a measurable map. Let...

Michelle Mendoza

Michelle Mendoza

Answered

2022-07-01

Let be some measurable domain and f : E R a measurable map. Let B Bor ( R ) be a Borel set. Show that f 1 ( B ) is measurable.
I'm advised to define A = { A f 1 ( A )  measurable }. Now A consists of sets whose preimage is measurable, and since f is continuous these sets are open. This collection seems to form a σ-algebra on R , but I'm confused about the construction here as it seems that A is the smallest σ-algebra containing open sets, but that would mean that it's equal to Bor ( R )?

Answer & Explanation

Freddy Doyle

Freddy Doyle

Expert

2022-07-02Added 20 answers

Your proof will depend on your definition of "measurable function".
Let's say "measurable function" means f 1 ( G ) is measurable for all open sets G R .
Then define A = { A R f 1 ( A )  measurable }. Show that A contains all open sets and that A is a sigma-algebra. Conclude that A Bor ( R ).
Pattab

Pattab

Expert

2022-07-03Added 2 answers

Note: you cannot prove A = Bor ( R ).

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