${Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right) dt $
Let's suppose for the purpose of the question x=1 so we observe : ${Ai}(1) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + t\right) dt $
I would say that , $\int_0^\infty f^+ = \infty$ and $-\int_0^\infty f^- = -\infty$ so that , the airy function is not defined in the lebesgue sense , but is a Riemann generalized integral. Is it a correct and complete argument ?