I have a measure μ that is supported on [ − 3 , 3 ] × R . What we are given is that, if we fix the first component i, then μ ( i , ⋅ ) is a probability measure. Formally (maybe it should integrate to d i or similar, I cannot define this very well): ∫ x ∈ R d μ ( i , x ) = 1 ∀ i ∈ [ − 3 , 3 ] .I find it very hard to understand this tuple-indexing notation. My question is the following: What kind of assumptions do we need to have a result similar to: ∫ ( i , x ) ∈ [ − 3 , 3 ] × R d μ ( i , x ) = ∫ i ∈ [ − 3 , 3 ] d iI just think that since for any fixed i the measure μ integrates to 1 (on the second dimension -- apologies for my poor terminology), then integrating over all ( i , x ) ∈ [ − 3 , 3 ] × R should also give simply an 'iterated integral' where we first integrate wrt x ∈ R for a fixed i, which will integrate to 1 , and then integrate over i ∈ [ − 3 , 3 ]. But of course, we cannot define ∫ ( i , x ) ∈ [ − 3 , 3 ] × R d μ ( i , x ) = ∫ i ∈ [ − 3 , 3 ] ∫ x ∈ R d μ ( i , x ) = ∫ i ∈ [ − 3 , 3 ] d y where in the red parts I make an abuse of integartion rules.

Question

I have a measure   μ that is supported on   [  −  3  ,  3  ]  ×      R  . What we are given is that, if we fix the first component   i, then   μ  (  i  ,  ⋅  ) is a probability measure. Formally (maybe it should integrate to       d    i or similar, I cannot define this very well):      ∫          x      ∈              R                  d    μ  (  i  ,  x  )  =  1    ∀  i  ∈  [  −  3  ,  3  ]  .I find it very hard to understand this tuple-indexing notation. My question is the following: What kind of assumptions do we need to have a result similar to:      ∫          (      i      ,      x      )      ∈      [      −      3      ,      3      ]      ×              R                  d    μ  (  i  ,  x  )  =      ∫          i      ∈      [      −      3      ,      3      ]            d    iI just think that since for any fixed   i the measure   μ integrates to 1 (on the second dimension -- apologies for my poor terminology), then integrating over all   (  i  ,  x  )  ∈  [  −  3  ,  3  ]  ×      R   should also give simply an &#039;iterated integral&#039; where we first integrate wrt   x  ∈      R   for a fixed   i, which will integrate to 1 , and then integrate over   i  ∈  [  −  3  ,  3  ]. But of course, we cannot define      ∫          (      i      ,      x      )      ∈      [      −      3      ,      3      ]      ×              R                  d    μ  (  i  ,  x  )  =            ∫              i        ∈        [        −        3        ,        3        ]                    ∫              x        ∈                  R                            d        μ    (    i    ,    x    )    =          ∫              i        ∈        [        −        3        ,        3        ]                    d        y  where in the red parts I make an abuse of integartion rules.

arhaitategr · Accepted Answer

The first equation you wrote, I think you probably mean   μ  (  {  i  }  ×      R    )  =  1?What you are trying to describe with iterated integrals might be the product measure and the Fubini/Tonelli theorems which allow you to convert an integral over a product measure       ν    1    ×      ν    2   to an iterated integral.

Finley Mckinney · Accepted Answer

∫          [      −      3      ,      3      ]      ×              R              d  μ  (  i  ,  x  )  =  μ  (  [  −  3  ,  3  ]  ×      R    ) by definition.

I have a measure μ that is supported on [ − 3 ,

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