I have difficulty understanding the some of the bounds given in the proof of Fubini's Theorem given by Stein in his text Fourier Analysis.
Suppose $f$ is continuous on $\mathbb{R}^2$ and of moderate decrease. Then $$F(x_1)=\int_{\mathbb{R}}f(x_1,x_2)dx_2$$ is of moderate decrease on $\mathbb{R}$, and $$\int_{\mathbb{R}^2}f(x)dx=\int_{\mathbb{R}}\Big(\int_{\mathbb{R}}f(x_1,x_2)dx_2\Big)dx_1.$$
In the proof below, I don't know how to get the first and second inequalities bounding $\Big|\int_{I_N}\Big(\int_{I_{N^c}}f(x_1,x_2)dx_2\Big)dx_1\Big|$ and the bound $\frac{C}{N}$ on $\Big|\int_{I_{N^c}}\Big(\int_{\mathbb{R}}f(x_1,x_2)dx_2\Big)dx_1\Big|$.
I would greatly appreciate it if anyone could show how to attain these bounds.
Observe \begin{align} &\left|\int_{I_N} \left(\int_{I_N^C}f(x_1, x_2) \ dx_2\right)dx_1\right| \leq \int_{I_N}\int_{I_N^c}|f(x_1, x_2)|\ dx_2 dx_1\\ &\leq \int_{0\leq |x_1| \leq 1}\int_{I_N^c}|f(x_1, x_2)|\ dx_2 dx_1+ \int_{1\leq |x_1| \leq N} \int_{I_N^c}|f(x_1, x_2)|\ dx_2 dx_1\\ &=: I_1 +I_2. \end{align}
For $I_1$, using the definition of moderate decreasing function, we have that \begin{align} I_1 \leq& \int_{0\leq |x_1| \leq 1} \int_{|x_2|\geq N} \frac{A}{1+(x_1^2+x_2^2)^{3/2}}\ dx_2 dx_1 \leq \int_{0\leq |x_1| \leq 1}\int_{|x_2|\geq N} \frac{A}{1+|x_2|^3}\ dx_2dx_1\\ \leq&\ \int_{0\leq |x_1| \leq 1}\int_{|x_2|\geq N} \frac{A}{|x_2|^3}\ dx_2dx_1 \leq 2A \int^1_{-1} \int_N^\infty \frac{1}{x_2^3}\ dx_2dx_1 = A' \int^1_{-1} \frac{1}{N^2}\ dx_1 = \frac{A''}{N^2}= \mathcal{O}\left(\frac{1}{N^2}\right). \end{align} For $I_2$, we are essentially given the same argument. Observe \begin{align} I_2 \leq& \int_{1\leq |x_1| \leq N} \int_{|x_2|\geq N} \frac{A}{1+(x_1^2+x_2^2)^{3/2}}\ dx_2dx_1 \leq \int_{1\leq |x_1| \leq N}\int_{|x_2|\geq N} \frac{A}{|x_2|^3}\ dx_2dx_1 \\ \leq& 4A \int^N_{1} \int_N^\infty \frac{1}{x_2^3}\ dx_2dx_1 = A' \int^N_{1} \frac{1}{N^2}\ dx_1 \leq \frac{A''}{N}= \mathcal{O}\left(\frac{1}{N}\right). \end{align}
In the case where we want to bound \begin{align} \left|\int_{I_N^c} \int_\mathbb{R} f(x_1, x_2)\ dx_2dx_1 \right| \end{align} the trick is similar. Observe \begin{align} \left|\int_{I_N^c} \int_\mathbb{R} f(x_1, x_2)\ dx_2dx_1 \right|\leq \ \left|\int_{I_N^c} \int_{I_N} f(x_1, x_2)\ dx_2dx_1 \right|+ \left|\int_{I_N^c} \int_{I_N^c} f(x_1, x_2)\ dx_2dx_1 \right|=: J_1 + J_2. \end{align}
For $J_1$, we have that \begin{align} J_1 \leq& \int_{I_N^c} \int_{I_N} |f(x_1, x_2)| \ dx_1dx_2 \leq 4\int^\infty_N\int^N_0 \frac{A}{1+(x_1^2+x_2^2)^{3/2}}\ dx_2dx_1\\ \leq&\ A' \int_N^\infty \int^N_0 \frac{1}{1+x_1^3}\ dx_2dx_1 = A' \int_N^\infty \frac{N}{1+x_1^3}\ dx_1 \leq A'N \int^\infty_N \frac{1}{x_1^3}\ dx_1 \leq A'' \frac{N}{N^2} = \mathcal{O}\left(\frac{1}{N} \right). \end{align} Likewise, we have \begin{align} J_2 \leq& \int_{I_N^c} \int_{I_N^c} |f(x_1, x_2)| \ dx_1dx_2 \leq 4 \int^\infty_N \int^\infty_N \frac{A}{1+(x_1^2+x_2^2)^{3/2}}\ dx_2dx_1\\ \leq&\ \int^\infty_N \int^\infty_N \frac{A'}{(x_1^2+x_2^2)^{3/2}}\ dx_2dx_1. \end{align} Using the fact that \begin{align} (x^2_1+x_2^2)^{3/2} \geq |x_1|^3+|x_2|^3 \end{align} we get that \begin{align} J_2 \leq& \int^\infty_N \int^\infty_N \frac{A'}{x_1^3+x_2^3}\ dx_2 dx_1 = \int^\infty_N \frac{A'}{x_1^3}\int^\infty_N \frac{1}{1+(\frac{x_2}{x_1})^3}\ dx_2dx_1 \\ =& \int^\infty_N \frac{A'}{x_1^2}\int^\infty_{N/x_1} \frac{1}{1+u^3}\ du dx_1 \leq \int^\infty_N \frac{A'}{x_1^2}\int^\infty_0\frac{1}{1+u^3}\ du dx_1\\ \leq&\ A''\int^\infty_N \frac{1}{x_1^2}\ dx_1 = \frac{C}{N}. \end{align}