Let g : R 0 + ⟶ ( 0 , 1 ) : x ↦ exp ⁡ ( − x ) be a function and let M = { M ⊆ R + ∣ M denumerable or M ¯ denumerable } and M ′ = { M ⊆ ( 0 , 1 ) ∣ M denumerable or M ¯ denumerable } be σ-Algebras .I am trying to show that the function g is not M − M ′ measurable, but I am missing the the counterexample. I gave it some thought and I think g should be M − B ( ( 0 , 1 ) ) measurable, because in M ′ are also Vitali sets included. Any help is much appreciated

Question

Let   g  :            R        0          +        ⟶  (  0  ,  1  )  :  x  ↦  exp  ⁡  (  −  x  ) be a function and let       M    =      {    M    ⊆                  R                    +              ∣    M       denumerable or             M      ¯       denumerable } and             M              ′        =  {  M  ⊆  (  0  ,  1  )  ∣  M denumerable or             M      ¯       denumerable } be   σ-Algebras .I am trying to show that the function g is not   M  −            M              ′       measurable, but I am missing the the counterexample. I gave it some thought and I think g should be   M  −      B    (  (  0  ,  1  )  ) measurable, because in             M              ′       are also Vitali sets included. Any help is much appreciated

aniizl · Accepted Answer

g  is a bijection so E is denumerable if and only if       g          −      1        (  E  ) is denumerable. Also       g          −      1        (      E          c        )  =  (      g          −      1        (  E  )      )          c      . Hence, f is       M    →            M        ′   measurable.[I am writing       A          c       for the complement of A].      g          −      1        (  0  ,      1    2    ) is not in       M   so g is not       M    →      B    (  0  ,  1  ) measurable.

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