Too strong assumption in the Uniqueness Theorem of Rudin's Real and Complex Analysis? In Rudin's Real and Complex Analysis, there is the following result about Fourier transforms.
The Uniqueness Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}(t)=0$ for all $t\in\mathbb{R}$, then $f(x)=0$ almost everywhere.
Isn't the assumption "for all $t\in\mathbb{R}$" unnecessarily too strong? I am pretty sure that we only need that $\hat{f}(t)=0$ almost everywhere to conclude that $f(x)=0$ almost everywhere. But Rudin is a very smart guy, so I guess there is a good reason for saying "for all $t\in\mathbb{R}$".
Note: In Rudin, the Uniqueness Theorem is a corollary of the following theorem.
The Inversion Theorem If $f\in L^1(\mathbb{R})$ and $\hat{f}\in L^1(\mathbb{R})$, and if
$$g(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \hat{f}(t)e^{ixt}dt\qquad(x\in\mathbb{R}),$$
then $g\in C_0$ and $f(x)=g(x)$ almost everywhere.

As Ian pointed out, the Fourier transform of an $L^1$ function is continuous. For continuous functions, being zero a.e. and being zero everywhere are equivalent; so it makes more sense to use the shorter version, without a.e. 
