Levy measure - $dx$ symbol

Question

Hey I have problem with understanding notation of levy measure. For example for Variance Gamma process most authors writes $$\Pi_{VG}\left(dx\right)=C\frac{e^{-Mx}}{x}1_{\left\{ x>0\right\} }dx+C\frac{e^{Gx}}{\left\vert x\right\vert }1_{\left\{ x<0\right\} }dx$$ but what it is $dx$? Measure works on sets. Can anyone explain how to go from Levy measure to Levy density?

Answer 1

$$\Pi_{VG}\left(dx\right)=C\frac{e^{-Mx}}{x}1_{\left\{ x>0\right\} }dx+C\frac{e^{Gx}}{\left\vert x\right\vert }1_{\left\{ x<0\right\} }dx$$ means $$\Pi_{VG}\left(E\right)=C\int_E \frac{e^{-Mx}}{x}1_{\left\{ x>0\right\} }dx+C\int_E\frac{e^{Gx}}{\left\vert x\right\vert }1_{\left\{ x<0\right\} }dx$$ for all $E$ where $\int f(x)dx$ is integration w.r.t. Lebesgue measure.

Answer 2

In probability theory, the following notation is commonly used:

Let $f \geq 0$ be a function. Then if one writes $\nu = f dx$ where $\nu$ is a measurue, it means that $$\nu(A) = \int_A f dx$$ In your situation, $$\Pi_{VG}(A) = \int_A C \frac{e^{-Mx}}{x} 1_{\{x > 0\}} dx + \int_A C \frac{e^{Gx}}{|x|} 1_{\{x < 0\}} dx$$