I have this question that's bothering me. The answer (I think) should be trivial but I am new in measure theory.Let m be Radon measure on a compact product space Ω = X × X such that m is symmetric (I guess this means that m ( d x , d y ) = m ( d y , d x ) and let λ be a symmetric strictly positive function λ : X × X → R . The question is the following: does it always exist a measure μ defined on X such that μ ( d x ) μ ( d y ) = c m ( d x , d y ) λ ( x , y ) ,where c is a positive constant? How do you find it? Is it possible to say that it is a probability measure?My immediate answer would be yes and μ ( d x ) = ∫ m ( ⋅ , d y ) λ ( x , y ) and it is a probability measure just because it can be rescalated (being X a compact set)Am I wrong? If so, can you give me a counterexample?

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I have this question that&#039;s bothering me. The answer (I think) should be trivial but I am new in measure theory.Let   m be Radon measure on a compact product space   Ω  =  X  ×  X such that   m is symmetric (I guess this means that   m  (  d  x  ,  d  y  )  =  m  (  d  y  ,  d  x  ) and let   λ be a symmetric strictly positive function   λ  :  X  ×  X  →      R  . The question is the following: does it always exist a measure   μ defined on   X such that  μ  (  d  x  )  μ  (  d  y  )  =  c            m      (      d      x      ,      d      y      )              λ      (      x      ,      y      )        ,where c is a positive constant? How do you find it? Is it possible to say that it is a probability measure?My immediate answer would be yes and  μ  (  d  x  )  =  ∫            m      (      ⋅      ,      d      y      )              λ      (      x      ,      y      )      and it is a probability measure just because it can be rescalated (being   X a compact set)Am I wrong? If so, can you give me a counterexample?

komizmtk · Accepted Answer

No. Let   X  =  [  0  ,  1  ],   m be the uniform distribution on the diagonal   D  =  {  (  x  ,  x  )  ∣  x  ∈  [  0  ,  1  ]  }, and let   λ have the constant value 1. The question then reduces to whether the uniform distribution on the diagonal is proportional to a product measure. It is not. Indeed, the marginal distributions of   m are the uniform distribution on [0,1], but the corresponding product measure assigns measure zero to the diagonal.

I have this question that's bothering me. The answer (I think) should be trivial but I am new in mea

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