I'm reading this paper which relates the sensitivity conjecture to a graph-theoretic question on the hypercube and I'm struggling to understand some things. A Boolean function here is defined as $f: \{\pm 1\}^n \to \{\pm 1\}$. $\mathbb{E}(f) = 2^{-n}\sum_{x \in V(Q_n)} f(x)$. The authors claim (without proof) that statements 1 and 2 are equivalent to 1' and 2' respectively.
My doubts are the following:
a) I can see that 1 $\iff$ 1' if $\rm{deg}_G (x) = n - s(g, x)$, and $\mathbb{E}(g) \neq 0 \iff |V(G)| \neq 2^{n-1}$. These hold for the characteristic Boolean function $g(x) = 1 \text{ iff } x \in V(G)$ which they have defined just before, but the statement 1' is claimed for any Boolean function $g$. They use a different $g$ when they prove that 1' $\iff$ 2' later. I'm a little confused here. Does the first condition hold for any Boolean function $g$ with $\mathbb{E}(g) \neq 0$? Is the characteristic function just mentioned as an easy example? That seems a bit odd.
b) This is a minor point, but I think in the statement of Theorem 2.1 they should mention the function $h$ needs to be invertible and crucially $h^{-1}$ is increasing, which is needed to show $2 \iff 2'$.
