I learned the following from Hunter's Applied Analysis. Denote the Schwartz space $${\mathcal S}({\mathbb R}^n):=\{\varphi\in C^{\infty}({\mathbb R}^n):\sup_{x\in{\mathbb R}^n}|x^{\alpha}\partial^{\beta}\varphi(x)|<\infty\quad\text{for every}\quad\alpha,\beta\in{\mathbb Z}_+^n\}.$$ If $\varphi\in{\mathcal S}({\mathbb R}^n)$, then the Fourier transform $\hat{\varphi}:{\mathbb R}^n\to{\mathbb C}$ is the function defined by $$\hat{\varphi}(\xi):=\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb R}^n}\varphi(x)e^{-i\xi\cdot x}dx,\quad \xi\in{\mathbb R}^n.$$

Suppose that $f,\varphi\in{\mathcal S}$. Then we have \begin{align}\int\hat{f}(\xi)\varphi(\xi)d\xi&=\int\frac{1}{(2\pi)^{n/2}}\bigg(\int f(x)e^{-i\xi\cdot x}dx\bigg)\varphi(\xi)d\xi\\ &=\int f(x)\frac{1}{(2\pi)^{n/2}}\bigg(\int\varphi(\xi)e^{-i\xi\cdot x }d\xi\bigg)dx\\ &=\int f(x)\hat{\varphi}(x)dx. \end{align} This is the motivation of the definition of the Fourier transform of tempered distributions.

Here is my question:

How can I get the second equality by Fubini's theorem?

The form I know about the theorem is $$\int_{A}\bigg(\int_B f(x,y)dy\bigg)dx =\int_{B}\bigg(\int_A f(x,y)dx\bigg)dy =\int_{A\times B}f(x,y)d(x,y)$$ But I am wondering what is $f(x,y)$ in the case above.

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The trick is to move everything inside the integrals and what is there will be the thing to which you'll have to apply Fubini. So you'll apply it to $g(x,\xi) = \frac{1}{(2\pi)^{n/2}} f(x) \varphi(\xi) e^{-\xi \cdot x}$. – t.b. Jul 20 '11 at 2:09
@Theo: +1.Thanks for your answer! I focus too much on the "difference" to notice the trick. Could you write your comment as an answer? – Jack Jul 20 '11 at 2:13
Okay, will do. But I don't have much more to say. By the way, there is a $dx$ missing in the equation you ask about. – t.b. Jul 20 '11 at 2:15
@Theo: Fair enough. That's good enough to be an answer. – Jack Jul 20 '11 at 2:16

The function to which you'll have to apply Fubini is $$g(x,\xi) = \frac{1}{(2\pi)^{n/2}} f(x) \varphi(\xi) e^{-\xi \cdot x}.$$ I think you can check for yourself that this is a function in $\mathcal{S}(\mathbb{R}^{n}\times\mathbb{R}^{n}) \subset L^{1}(\mathbb{R}^n \times \mathbb{R}^n)$, so it is a function to which we can apply Fubini.