Definition of the Laplace Transform

Question

I'm studying from the book "Borel-Laplace Transform and Asymptotic Theory" by Sternin and Shatalov. At page 10, the authors provide the following definition of the Laplace Transform:

However, this definition isn't the same as the one I am used to, which reads as follows:

Is there any motivation or explanation of this? Thank you in advance.

Edit: The footnote 3 in the definition says