Let M := ( Ω , F ) be a measurable space and let M be the vector space of all ( R , B ( R ) )-valued measurable functions defined on M. If G is a sub- σ-field of F , let M G be the subspace of M whose elements are the G-measurable functions.Suppose that F 1 , … , F k are sub- σ-fields of F and let F := ∨ i = 1 k F i denote their join.Is there any relationship between M F and M F i besides that the latter is a subset of the former? Something like the former being the sum of the latter? ⨁ i = 1 k M F i ⊆ M F holds (where ⨁ i = 1 k M F i := { f 1 + … + f k : f i ∈ M F i , i ∈ { 1 , … , k } }, but I can't see how to prove the reverse inclusion if it is indeed true.

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Let   M  :=  (  Ω  ,      F    ) be a measurable space and let       M   be the vector space of all   (      R    ,      B    (      R    )  )-valued measurable functions defined on   M. If   G is a sub-  σ-field of       F  , let             M              G       be the subspace of       M   whose elements are the   G-measurable functions.Suppose that             F              1        ,  …  ,            F              k       are sub-  σ-fields of       F   and let       F    :=      ∨          i      =      1              k                  F              i       denote their join.Is there any relationship between             M                      F             and             M                                F                          i                     besides that the latter is a subset of the former? Something like the former being the sum of the latter?      ⨁          i      =      1              k                  M                                F                      i                                ⊆            M                      F             holds (where       ⨁          i      =      1              k                  M                                F                      i                                :=  {      f          1        +  …  +      f          k        :      f          i        ∈            M                                F                          i                      ,  i  ∈  {  1  ,  …  ,  k  }  }, but I can&#039;t see how to prove the reverse inclusion if it is indeed true.

Amir Beck · Accepted Answer

Consider the measure space   (  [  −  1  ,  1  ]  ,            B              [      −      1      ,      1      ]        ), and let             F        1    =  σ  (  [  0  ,  1  ]  ) and             F        2    =  σ  (  f  ), where   f  (  x  )  =      |    x      |  . Note that             F        2   consists of Borel sets in [−1,1] that are symmetric around 0, and             F        1    ∨            F        2   is generated by the sets in   {  A  ∩  B  :  A  ∈            F        1    ,  B  ∈            F        2    }. Therefore,       F    =            B              [      −      1      ,      1      ]      . Now, the identity function is measurable w.r.t.       F  , but it cannot be represented as a sum of             F        1   and             F        2   measurable functions.

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