I am reading the following from Tao: https://www.math.ucla.edu/~tao/247a.1.06f/notes4.pdf ;
On page 5, it says observe that if $f \in L^2(\mathbb{R})$ and $y$ lies outside of the support of $f$, then $$ Hf(y) = \int_{\mathbb{R}} \frac{1}{\pi (y-x)} f(x)dx $$
I am not sure how to rigorously justify this statement, and in particular how to take advantage of the compact support in order to use something like DCT/MCT. In addition, does this holds true for general $L^p$ spaces?
We can write $f$ as $f= f\cdot\chi_K$, where $\chi_K$ is the characteristic function for the set $K$, and $K$ is the support of $f$. Next, we apply some inequality from our toolbox in order to estimate the integral $$\int_{\mathbb{R}}\frac{\chi_K(x)}{y-x}f(x)dx$$ which I leave to you (but please say if you get stuck).
Also: Why must $K$ be compact?