Vrftjkry

Let $ngeq 2$ and consider the improper integral
$$I:=int_{mathbb{R}^{n}}F(x)dx$$ where $F$ is a continuous function.

If $I$ exists then

$$I=lim_{Rrightarrow +infty}int_{B_{R}}F(x)dx,$$
where $B_{R}$ is a ball with radius $R$.
So if this limit does not exist we know that the integral does not exist.
Does the existence of this limit imply the existence of the integral ?

Motivation:

I am interested in the existence of the integral
$$int_{mathbb{R}^{3}}frac{e^{dot{imath}|x-y|^2}}{1+|y|}dy.$$

Using spherical coordinates (I do not even know if we are allowed to change variables here. Are we ? )
$$int_{mathbb{R}^{3}}frac{e^{dot{imath}|x-y|^2}}{1+|y|}dy=
int_{mathbb{S}^{2}}int_{0}^{infty}
frac{e^{dot{imath}|rhoomega-x|^2}rho^2}{1+rho}drho domega\
=e^{i|x|^{2}}int_{mathbb{S}^{2}}int_{0}^{infty}
frac{e^{dot{imath} (rho^2-2xcdot omega,rho)}rho^2}{1+rho}drho domega.$$

Observations:

1-The inner integral does not exist for any $x$ and $omega$.

2-We can not change order of integration

3-The limit

$$lim_{Rrightarrow infty}int_{mathbb{S}^{2}}int_{0}^{R}
frac{e^{dot{imath} (rho^2-2xcdot omega,rho)}rho^2}{1+rho}drho domega$$
exists. Simply apply the very nice formula [Grafakos, classical Fourier analysis-Appendix D]:
$$int_{mathbb{S}^{n-1}}
F(x.omega)domega=c int_{-1}^{1}(sqrt{1-s^2})^{n-3}
F(s|x|)ds.$$
then benefit from the oscillation in both variables $rho$ and $omega$ and integrate by parts in both variables.

Any ideas how to handle this ?

Thank you so much

Does the existence of this limit imply the existence of the integral ?

No. Take $n=1$. Then $$lim_{Rtoinfty}int_{-R}^{R}x,dx$$ exists and equals $0$ because postive parts are exactly canceling negative parts as $R$ grows. But I would not say $int_{mathbb{R}}x,dx$ exists.

Maybe there is a question yet to answer if you require the function to be positive.