Change from stochastic exponential to exponential of Lévy process - Applebaum

Question

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process.

I am not sure about the definition of $\nu\circ f^{-1}$. I mean $$\nu_1(dx)=?$$
Thanks a lot!

Answer 1

$\nu \circ f^{-1}$ denotes the push-forward measure (or image measure) of $f$ with respect to $\nu$, i.e.

$$\nu_1(B) = (\nu \circ f^{-1})(B) = \int 1_B(f(x)) \, \nu(dx)$$

for any Borel set $B$. This implies

$$\int g(x) \, \nu_1(dx) = \int g(f(x)) \, \nu(dx)$$

for any $g \in L^1(\nu_1)$.