Let (X,M), (Y,N) and (Z,O) be measurable spaces, and let f:X->Y and g:Y->Z be measurable functions. Then sigma(g∘f)=sigmaσ(f) if and only if g is bijective, where sigma(g∘f)=(g∘f)−1(O)=f−1(g−1(O)) and σ(f)=f−1(N).

Cael Dickerson

Cael Dickerson

Answered question

2022-11-07

Let ( X , M ), ( Y , N ) and ( Z , O ) be measurable spaces, and let f : X Y and g : Y Z be measurable functions. Then σ ( g f ) = σ ( f ) if and only if g is bijective, where σ ( g f ) = ( g f ) 1 ( O ) = f 1 ( g 1 ( O ) ) and σ ( f ) = f 1 ( N ).

Answer & Explanation

Quinten Cervantes

Quinten Cervantes

Beginner2022-11-08Added 13 answers

A measurable bijection between standard Borel spaces is a Borel isomorphism. Therefore,
σ ( T ( X ) ) = X 1 ( T 1 ( B ( R m ) ) ) = X 1 ( B ( R n ) ) = σ ( X ) .

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