Consider $\mathbb{R}^2$ with Lebesgue measure and let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a measurable function. Assume that $$\iint f(x,y) dx dy < \infty$$ Is $f$ integrable? Suppose $f$ is integrable, is then true that $$\int_{\mathbb{R}^2} f = \iint f(x,y)dx dy\, ?$$
I know that if the on $[0,1]^2$ the answer for the first question is negative considering $$f(x,y) = \frac{x^2-y^2}{(x^2+y^2)^2}$$ since both iterates are finite but $$\iint \left | \frac{x^2-y^2}{(x^2+y^2)^2}\right | dx dy = \infty$$ and hence it is not integrable. But I'm not sure if this works for $\mathbb{R}^2$ instead.
For the second question, I believe it is true by Fubini's theorem but I'd like to be sure. Thank you in advance!