Measurable and non-singular equivalence relation in a one-sided Markov ShiftIn a one-sided Markov shift ( X , B , μ , T ) where X = { 1 , 2 , ⋯ , n } N is the sample space, B the σ-algebra generated by finite cylinder sets, μ a product measure and T the shift transformation, first, we define a metric d ( x , y ) = 2 − n where n is the least integer such that x [ n ] ≠ y [ n ]. Then, we define the following relation:Given x , y ∈ X, x ∼ r y iff there exists n , m ∈ Z ≥ 0 such that T n x = T m y.One can easily check ∼ r is an equivalent relation. We set R = { ( x , y ) ∈ X × X | x ∼ r y } and, for each subset A ⊆ X, we set R ( A ) = ⋃ a ∈ A [ a ] ∼ r to be the union of equivalent classes of each element in A.My question is:1. Is R ∈ B × B = { A × B | A , B ∈ B }?2. For each B ∈ B , is it true that μ ( B ) = 0 iff μ [ R ( B ) ] = 0?The first question is asking if R is measurable and the second is asking if R is non-singular. For the first one, it suffices to show that, for each B ∈ B , R ( B ) ∈ B . However, R(B) could be properly contained in ⋃ n ∈ N T n ( B ) and then I do not know how to proceed. I also try to consider the function ( x , y ) ↦ inf n , m d ( T n x , T m y ). This function is continuous but it may obtain zero outside of R.For the second question, I suppose whenever μ ( B ) &gt; 0, B will contain a finite cylinder set and hence so does R(B). However, I do not how to deal with the other direction.

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Measurable and non-singular equivalence relation in a one-sided Markov ShiftIn a one-sided Markov shift   (  X  ,      B    ,  μ  ,  T  ) where   X  =  {  1  ,  2  ,  ⋯  ,  n      }                  N             is the sample space, B the   σ-algebra generated by finite cylinder sets,   μ a product measure and T the shift transformation, first, we define a metric   d  (  x  ,  y  )  =      2          −      n       where n is the least integer such that   x  [  n  ]  ≠  y  [  n  ]. Then, we define the following relation:Given   x  ,  y  ∈  X,   x      ∼    r    y iff there exists   n  ,  m  ∈            Z              ≥      0       such that       T    n    x  =      T    m    y.One can easily check       ∼    r   is an equivalent relation. We set   R  =  {  (  x  ,  y  )  ∈  X  ×  X    |    x      ∼    r    y  } and, for each subset   A  ⊆  X, we set   R  (  A  )  =      ⋃          a      ∈      A        [  a      ]                  ∼        r             to be the union of equivalent classes of each element in A.My question is:1. Is   R  ∈      B    ×      B    =  {  A  ×  B    |    A  ,  B  ∈      B    }?2. For each   B  ∈      B  , is it true that   μ  (  B  )  =  0 iff   μ  [  R  (  B  )  ]  =  0?The first question is asking if R is measurable and the second is asking if R is non-singular. For the first one, it suffices to show that, for each   B  ∈      B    ,  R  (  B  )  ∈      B  . However, R(B) could be properly contained in       ⋃          n      ∈              N                  T    n    (  B  ) and then I do not know how to proceed. I also try to consider the function   (  x  ,  y  )  ↦      inf          n      ,      m        d  (      T    n    x  ,      T    m    y  ). This function is continuous but it may obtain zero outside of R.For the second question, I suppose whenever   μ  (  B  )  &amp;gt;  0, B will contain a finite cylinder set and hence so does R(B). However, I do not how to deal with the other direction.

Marshall Flowers · Accepted Answer

Step 1For the first question, the crucial fact is that R is the union of the sets      R          n      ,      m        =            {        (  x  ,  y  )  ∈  X  ×  X  :      T    n    (  x  )  =      T    m    (  x  )            }        ,as n and m range in N, each of which is closed in the product topology, and hence measurable. Consequently R is an       F    σ   set, therefore measurable.Step 2For the second question, notice that if B has measure zero, then so does T(B), and hence also       T    m    (  B  ), for every m. Moreover, since T preserves   μ (meaning that   μ  (      T          −      1        (  E  )  )  =  μ  (  E  ) for every measurable set E), we have that  μ  (      T          −      n        (      T    m    (  B  )  )  )  =  μ  (      T    m    (  B  )  )  =  0  ,for every n. The fact that R(B) has measure zero then follows from  R  (  B  )  =      ⋃          n      ,      m      ∈                        N                          T          −      n        (      T    m    (  B  )  )  ,

In a one-sided Markov shift (X,B, mu,T) where X={1,2,⋯,n}^N is the sample space, B the sigma-algebra generated by finite cylinder sets, mu a product measure and T the shift transformation, first, we define a metric d(x,y)=2^{-n} where n is the least integer such that x[n] ne y[n].

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