 Solution

The Fubini theorem for an integrable Borel-measurable function states that, if ${\int }_{{\mathbb{R}}^2}|f|<\infty$, its sections are integrable for almost every coordinate andFor a nonnegative measurable function, Tonelli theorem gives the same equalities with infinity allowed before integrability is established.

Here $f$ is the continuous exponential multiplied by the indicator of a Borel set, so it is Borel-measurable and nonnegative. Tonelli and integration in $x$ giveThis proves integrability. We may now integrate in the reverse order; for $x>0$ the inner integral is $(e^{-ax}-e^{-bx})/x$. Hence the Frullani integral isThe apparent singularity at zero is removable in the integrand's limiting value $b-a$.