Can one find an $f$ on $\mathbb{R}²$ which is integral-undefined on $\mathbb{R}²$ , but $$\int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y) {\rm d}x{\rm d}y=\int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y){\rm d}y{\rm d}x<∞$$
The integrals mentioned in here are all referring to the Lebesgue Integrals, and the integral-undeined means $\int f^+,\int f^-$ are both infinity.
I asked this question because I am learning the Fubini-Toneli's theorem, which states as:
if $f(x,y)$ is integral-defined on $\mathbb{R}^d$ , then for a.e $x\in\mathbb{R}^{d_1},y\in\mathbb{R}^{d_2}$,there holds $$\int_{\mathbb{R}^{d_2}}\int_{\mathbb{R}^{d_1}}f(x,y) {\rm d}x{\rm d}y=\int_{\mathbb{R}^{d_1}}\int_{\mathbb{R}^{d_2}}f(x,y){\rm d}y{\rm d}x=\int_{\mathbb{R}^d}f(x,y){\rm d}(x,y)$$
And I just wanna to know whether the converse statement is true.
Here is an example from Rudin's Real and Complex Analysis, 3rd edition.