Let's say I have a family of Borel measures ( μ t ) t ∈ [ 0 , 1 ] over R d such that the map t → μ t ( B ) is borel measurable for each Borel set B, and let's say I want to associate to this family a finite Borel measure μ over R d × [ 0 , 1 ] clearly such that if f ∈ C 0 ∞ ( R d × [ 0 , 1 ] ) the following holds: ∫ R d × [ 0 , 1 ] f ( x , t ) d μ = ∫ 0 1 ∫ R d f ( x , t ) d μ t d t .Is this always possible?edit: the family μ t is a family of finite borel measures

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Let&#039;s say I have a family of Borel measures   (      μ    t        )          t      ∈      [      0      ,      1      ]       over       R    d   such that the map   t  →      μ    t    (  B  ) is borel measurable for each Borel set   B, and let&#039;s say I want to associate to this family a finite Borel measure   μ over       R    d    ×  [  0  ,  1  ] clearly such that if   f  ∈      C    0    ∞    (      R    d    ×  [  0  ,  1  ]  ) the following holds:      ∫                  R        d            ×      [      0      ,      1      ]        f  (  x  ,  t  )  d  μ  =      ∫    0    1        ∫                  R        d              f  (  x  ,  t  )  d      μ    t    d  t  .Is this always possible?edit: the family       μ    t   is a family of finite borel measures

Jordan Mcpherson · Accepted Answer

Yes, this is always possible under the conditions which you assumed, namely that   t  ↦      μ    t    (  B  ) is measurable for all measurable sets   B and that the       μ    t   are finite.The notion that I think you&#039;re specifically looking for is the product of transition kernels. Note that every measure is trivially a transition kernel which is constant in its first coordinate.

Let's say I have a family of Borel measures ( μ t </msub>

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