Let S , T , I be non-empty sets, for every i ∈ I let A i be a sigma-algebra on T, and let f : S → T. Is it true that f − 1 ( ⋂ i ∈ I A i ) = ⋂ i ∈ I f − 1 ( A i )?Remarks1. It is known that f − 1 ( ⋂ i ∈ I B i ) = ⋂ i ∈ I f − 1 ( B i ) if B i ⊆ T for every i ∈ I. But this result is not the same as what needs to be proved, since ⋂ i ∈ I B i ⊆ T, whereas ⋂ i ∈ I A i ⊆ P T.2. I was able to show that f − 1 ( ⋂ i ∈ I A i ) ⊆ ⋂ i ∈ I f − 1 ( A i ), so the question reduces to whether the converse containment is valid.

Question

Let   S  ,  T  ,  I be non-empty sets, for every   i  ∈  I let             A        i   be a sigma-algebra on   T, and let   f  :  S  →  T. Is it true that       f          −      1                  (            ⋂          i      ∈      I                  A        i              )        =      ⋂          i      ∈              I                  f          −      1        (            A        i    )?Remarks1. It is known that       f          −      1                  (            ⋂          i      ∈      I            B    i              )        =      ⋂          i      ∈              I                  f          −      1        (      B    i    ) if       B    i    ⊆  T for every   i  ∈  I. But this result is not the same as what needs to be proved, since       ⋂          i      ∈              I                  B    i    ⊆  T, whereas       ⋂          i      ∈              I                        A        i    ⊆      P    T.2. I was able to show that       f          −      1                  (            ⋂          i      ∈      I                  A        i              )        ⊆      ⋂          i      ∈              I                  f          −      1        (            A        i    ), so the question reduces to whether the converse containment is valid.

zepplinkid7yk · Accepted Answer

Not true. Let   Ω  =  [  0  ,  1  ] and let             A        1    =  {  ∅  ,  Ω  ,  A  ,      A          c        } and             A        2    =  {  ∅  ,  Ω  ,  B  ,      B          c        } where   A  =  [  0  ,      1    2    )  ,  B  =  [  0  ,      1    3    ). Then the left side is   {  ∅  ,  Ω  }. I leave it to you to find an   f for which       f          −      1        (  A  )  =      f          −      1        (  B  ) (  ≠  ∅  ,  ≠  Ω) so the right side contains       f          −      1        (  A  ).

Let S , T , I be non-empty sets, for every i ∈ I let

Answered question

Answer & Explanation

New Questions in Pre-Algebra