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 ).

Question

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    ).

Quinten Cervantes · Accepted Answer

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  )  .

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).

Answered question

Answer & Explanation

New Questions in College Statistics