Why Ampère’s Law holds for infinite currents?

I’m reading Griffiths proof of Ampère’s Law and in that proof he needs to show that :

$$oint frac{(x – x^{‘})}{imath ^{3}}mathbf J cdot mathbf {da^{‘}} = 0 $$ where $$imath = sqrt {(x – x^{‘})^2 + (y – y^{‘})^2 + (z – z^{‘})^2} $$

He makes the claim that if $mathbf J$ is confined to finite space then $mathbf J_{surface} = 0$ and the integral indeed equals zero.

In the notes he writes : "If J itself extends to infinity (as in the case of an infinite straight wire), the surface integral is still typically zero, though the analysis calls for greater care."

My question is : Why is the surface integral still zero in the case where $mathbf J$ itself extends to infinity ?