I'm learning harmonic analysis with Grafakos' book Classical Fourier Analysis. I'm having a problem understanding the definition 5.2.1 of maximal singular integral operators.
Let me recall the definition. Here, $\Omega \colon \mathbb{S}^{n-1} \to \mathbb{C}$ is an integrable function with mean $0$. For $0 < \epsilon < N$ and $f \in L^p(\mathbb{R}^n)$ (for $1 \leq p < \infty$), we define the truncated singular integral $$ T^{(\varepsilon, N)} f(x) = \int_{\varepsilon \leq |y| \leq N} \frac{\Omega(y/|y|)}{|y|^n} f(x - y) \mathrm{d}y$$ This quantity is defined for almost all $x \in \mathbb{R}^n$, as one can easily prove that $\|T^{(\varepsilon, N)} f\|_p < \infty$. But as I understand it, the negligible set of points at which the above quantity is not defined depends on $\varepsilon$ and $N$. Therefore, I don't know how to make sense of the maximal singular operator $$ T^{(*,*)}f(x) = \sup_{\varepsilon, N} |T^{(\varepsilon, N)}f(x)| $$ Am I missing something?
Note that my concerns disappear in the case $\Omega$ is bounded, as in that case, $T^{(\varepsilon, N)}$ is defined for all $f$ and $x \in \mathbb{R}^n$.