# How the integration over $\mathbb{R}$ is transformed into summation over $\mathbb{R}_+$?

I'm not sure how this translation from integration (green box) to summation (blue box) occurred in the screenshot above. Furthermore, I'm also unsure about the change of $$\mathit{v}_k$$ into $$\mathit{v}(\text{du})$$.

The "Eqn (3b)" is referring to $$\underset{\mathbb{R}^d}{\int }(1\land |u|)\mathit{v}(\text{du})<\infty$$.

Please enlighten me in realizing how these transformations resulted.