Suppose $f:\mathbb{R}^2\to \mathbb{R}$ is such that $$\int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x,y)dy\right]dx =1$$ and $$\int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x,y)dx\right]dy =-1.$$ Let $m$ refers the Lebesgue measure on $\mathbb{R}^2$
What range of values can $$\iint_{\mathbb{R}^2} |f|dm$$ can possibly have?
I have very basic knowledge of measure theory. I would be like to have some hints. Thanks in advance. Efforts:
I looked up Fubini Theorem (https://en.wikipedia.org/wiki/Fubini%27s_theorem)
If I consider the contrapositive of Fubini's theorem, I conclude that $f$ is not measurable. I am stuck how to proceed from here.
Thanks again for reading.
If $\iint_{\mathbb R^2} |f| \, dm$ was a finite real number, then by Fubini's theorem
$$ \int_{\mathbb R} \left( \int_{\mathbb R} f(x,y) \, dx \right) \, dy = \int_{\mathbb R} \left( \int_{\mathbb R} f(x,y) \, dy \right) \, dx = \int_{\mathbb R^2} f \, dm. $$
We have that $\int_{\mathbb R} \left( \int_{\mathbb R} f(x,y) \, dx \right) \, dy \ne \int_{\mathbb R} \left( \int_{\mathbb R} f(x,y) \, dy \right) \, dx$, so $\iint_{\mathbb R^2} |f| \, dm$ must be $+\infty$. (This is just the contrapositive of Fubini's theorem.)