$L^1\big((1+x)\mathrm dx\big)$ form

Question

I am reading the article.
On page 15, they wrote

But I don't know why we can denote $\Phi(s) = \int_{0}^{+\infty}x^{s-1}\Phi(x)dx$.

For your convenience, I also attach the picture in which there are equation (5) and kernel $k$ form $(4)$.