Im trying to compute R(dx) from a paper by Jan Rosinski, which can be found here, (also here using slightly different notation).

In the paper on page 3 we have the following theorem

Theorem 2.3. The Levy measure M of a tempered alpha stable distribution can be written in the form

$\displaystyle M(A) = \int_{R^d}\int_0^{\infty} \textbf{\textif{I}}_A(tx)t^{-\alpha-1}e^{-t}dtR(dx)$

where I_A(tx) I assume is the indicator funtion, i.e. tx is defined on the interval A

now I'm using definition of the Gamma function kernal to say

$\displaystyle \int_0^{\infty} t^{-\alpha-1}e^{-t}dt = \Gamma(-\alpha)$

I know the Levy measure M, and so putting that in for M, I then had

$\displaystyle 2^{\alpha}\delta\frac{\alpha}{\Gamma(1-\alpha)}x^{-1-\alpha}e^{-0.5\gamma^{1/\alpha}x} = \Gamma(-\alpha)\int_{R^d}R(dx)$

So I need to work out what the function R(dx) is, which is my problem. I was thinking I could just differentiate both sides which would get the integral out of the right hand side and then I could easily rearrange to find R(dx). However I am not sure about this method as I have this integral over the range R^d.

How do I work out what R(dx) is?

I was told in another forum by someone...

"From the statement of the theorem, it seems that equation (2.6) holds for a given Borel measure satisfying the finiteness criterion (2.8). Is this characterization of the measure not enough for your computations?"

However I fail to see how this helps. Please any help in how I compute R(dx) would be much appreciated.