Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a radial limit for almost all $z\in\mathbb{T}$ the border of $\mathbb{D}$.

It is said that $q\in H^2$ is inner if $|q(z)|=1$ for almost all $z\in\mathbb{T}$.

Now, I have read the following lemma in Helson's book *Lectures on Invariant Spaces*:

**There exists $f\in H^2$ such that $q\cdot\overline{f}\notin H^2$ for all $q\in H^2$ inner.**

The proof relies on two facts:

- if $f\in H^2$ and $q\cdot\overline{f}\in H^2$ then the real part of $f$ must be non-zero almost everywhere in $\mathbb{T}$ (unless $f$ is identically zero),
- there exists $f\in H^2$ non identically zero such that its real part vanishes in a set of positive measure of $\mathbb{T}$.

I fail to see why any of these two facts hold and the book gives no further explanation either. Any ideas?