I am having some trouble understanding how to apply Fubini and/or Tonelli Theorems to determine whether a Lebesgue integral over $\mathbb{R}^2_+$ exists and if it is finite.
If someone could help me by showing the explicit steps for the examples below I would be grateful. I have a long list of exercises I have found online (this is self-study) and instead of posting a bunch of examples here I thought a few simple ones would help me learn how to do these problems going forward.
The examples in hand are: for each of the functions, use Fubini or Tonelli to show the existence/finiteness of the function's Lebesgue integral over $\mathbb{R}^2_+$.
$f_1(x,y)=\frac{\sin xy}{xy}$
$f_2(x,y)=e^{-(1+x^2)y}$
Many thanks in advance!
Assume $f_1$ is integrable in $\mathbb{R}^2_+$ (which I assume is $\mathbb{R}\times \mathbb{R}_+$), by Fubini's theorem $f_1^x(y)=f_1(x,y)$ would be (Lebesgue!) integrable for almost every $x\in \mathbb{R}$, but $$ \int_{\mathbb{R}_+} f_1^x(y) dy= \int_0^{\infty} \frac{\sin(xy)}{xy} dy $$ and this last is not a Lebesgue integral for $x\neq 0$ (the positive and negative parts of the function $\sin(z)/z$, when integrated give $\infty$). So $f_1$ is not integrable in $\mathbb{R}^2_+$.
Since $f_2$ is positive everywhere, Tonelli's theorem guarantees that we can integrate first over $y$ and then over $x$ to get $$ \int_{\mathbb{R}^2_+} f_2(x,y)d(x,y) = \int_{\mathbb{R}} \int_0^\infty e^{-(1+x^2)y}dydx= \int_{-\infty}^{\infty} \frac{1}{1+x^2} dx <\infty $$