Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be Borel-measurable. State Fubini's theorem for the double integral

$$\int_{y \in \mathbb{R}} \int_{x \in \mathbb{R}} f(x, y) d x d y$$

Let $0<a<b$. Show that the function

$$f(x, y)= \begin{cases}e^{-x y} & \text { if } x \in(0, \infty), y \in[a, b] \\ 0 & \text { otherwise }\end{cases}$$

is measurable and integrable on $\mathbb{R}^{2}$.

Evaluate

$$\int_{0}^{\infty} \frac{e^{-a x}-e^{-b x}}{x} d x$$

by Fubini's theorem or otherwise.