In the context of the Lévy-Khintchine formula, I have certain integral (1) ∫ R p ( e i t ′ x − 1 − i t ′ x 1 − x ′ x ) d ν ( x ) , t ∈ R p where ν is a measure defined on R p .Suppose that ∫ R p | x | 2 d ν ( x ) &lt; ∞Define: κ ( E ) = ∫ E | x | 2 d ν ( x ) , E ⊂ R p .I want to get d κ ( x ) as a function of d ν ( x ) in order to do a substitution in (1), but I don't know how to differentiate κ: d κ ( x ) = ? d ν ( x )

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In the context of the Lévy-Khintchine formula, I have certain integral                    (1)                              ∫                                                    R                            p                                                            (                                    e                      i                          t              ′                        x                          −        1        −        i                                            t              ′                        x                                1            −                          x              ′                        x                                                )                          d        ν        (        x        )        ,                t        ∈                              R                    p                    where   ν is a measure defined on             R        p  .Suppose that      ∫                            R                p                  |    x            |        2    d  ν  (  x  )  &amp;lt;  ∞Define:  κ  (  E  )  =      ∫          E            |    x            |        2    d  ν  (  x  )  ,    E  ⊂            R        p    .I want to get   d  κ  (  x  ) as a function of   d  ν  (  x  ) in order to do a substitution in (1), but I don&#039;t know how to differentiate   κ:  d  κ  (  x  )  =  ?  d  ν  (  x  )

humusen6p · Accepted Answer

The Radon-Nikodym derivative.If   κ  (  E  )  =      ∫          E            |    x            |        2    d  ν  (  x  ) for all measurable   E  ⊆            R        p  , then       |    x            |        2   is called the Radon-Nikodym derivative of   κ with respect to   ν. Some notations are      |    x            |        2    =            d      κ              d      ν        (  x  )    or        |    x            |        2      d  ν  (  x  )  =      d    κ    (  x  )  .The meaning of each of these is:  ∫  φ  (  x  )    d  κ  (  x  )  =  ∫  φ  (  x  )        |    x            |        2      d  ν  (  x  )for all appropriate functions   φ.

In the context of the Lévy-Khintchine formula, I have certain integral <mtable displaystyle="tr

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