Let ν denote counting measure and λ denote Lebesgue measure. What is ( ν ⊗ λ ) ( { ( x , x ) } ) x ∈ R ?I am a little fuzzy on the product measure and have tried two ways to do this. Can someone explain why the incorrect version is wrong and the correct version is right?By definition of the product measure ( ν ⊗ λ ) ( { ( x , x ) } ) x ∈ R = ν ( { ( x , x ) } x ∈ R ) λ ( { ( x , x ) } x ∈ R ) = 0 since each λ ( { ( x , x ) } ) = 0 and ( ν ⊗ λ ) ( { ( x , x ) } ) x ∈ R = ∫ λ ( { x } ) ( χ { ( x , x ) } ) y d μ = 0 and ( ν ⊗ λ ) ( { ( x , x ) } ) x ∈ R = ∫ μ ( { x } ) ( χ { ( x , x ) } ) x d λ = ∞. So this value is not well defined. If any of these are right, I would like to know why one is right, the other is wrong. If they are both wrong, I would appreciate an explanation on how to do this properly.

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Let       ν   denote counting measure and       λ   denote Lebesgue measure. What is   (      ν    ⊗      λ    )  (  {  (  x  ,  x  )  }      )          x      ∈              R            ?I am a little fuzzy on the product measure and have tried two ways to do this. Can someone explain why the incorrect version is wrong and the correct version is right?By definition of the product measure   (      ν    ⊗      λ    )  (  {  (  x  ,  x  )  }      )          x      ∈              R              =  ν  (  {  (  x  ,  x  )      }          x      ∈              R              )  λ  (  {  (  x  ,  x  )      }          x      ∈              R              )  =  0 since each   λ  (  {  (  x  ,  x  )  }  )  =  0 and  (      ν    ⊗      λ    )  (  {  (  x  ,  x  )  }      )          x      ∈              R              =  ∫  λ  (  {  x  }  )  (      χ          {      (      x      ,      x      )      }            )    y    d  μ  =  0 and   (      ν    ⊗      λ    )  (  {  (  x  ,  x  )  }      )          x      ∈              R              =  ∫  μ  (  {  x  }  )  (      χ          {      (      x      ,      x      )      }            )    x    d  λ  =  ∞. So this value is not well defined. If any of these are right, I would like to know why one is right, the other is wrong. If they are both wrong, I would appreciate an explanation on how to do this properly.

lorienoldf7 · Accepted Answer

All three are wrong. The first one doesn&#039;t make sense. The second and third are both trying to use Tonelli&#039;s theorem, but Tonelli&#039;s theorem does not hold here since the counting measure is not sigma-finite. A correct way to proceed is to use the definition of the product measure as the inf of sum of measures of measurable rectangles that cover your set. Assume that we use the Borel sigma algebra on       R  . You can show that   (  μ  ×  ν  )  (  {  (  x  ,  x  )  :  x  ∈  [  0  ,  1  ]  }  )  =  ∞. The case of   μ and   ν on the Borel subsets of [0,1] is a classic example of when all three things that are supposed to be equal in Tonelli&#039;s theorem are unequal.

tr2os8x · Accepted Answer

You are correct. In order to uniquely define the product measure, you need the measures be   σ-finite in their respective spaces. In particular, the counting measure is not   σ-finite in       R  .

Let <mrow class="MJX-TeXAtom-ORD"> ν </mrow> denote counting measure and <mrow cl

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