Start with a set $X={1,2,...,n}$ of $n$ elements and the family $2^X$ of all subsets of $X$. For a real number $p$ between zero and one, we consider a probability distribution $\mu_p$ on $2^X$ where the probability that $i \in S$ is $p$, independently for different $i$'s. Thus for $p=1/2$ we get the uniform probability distribution.

For $S \subseteq {0, \ldots, n-1}$, its probability is $p^{|S|} (1-p)^{n - |S|}$.

For $p = 1/2$, the $p$-biased measure is the uniform distribution: $\mu_{1/2}(S) = (1/2)^n$ for every $S \subseteq [n]$.

The $p$-biased measure of a family $\mathcal F \subseteq 2^{[n]}$, i.e. $\mu_p(\mathcal F) = \sum_{S \in \mathcal F} \mu_p(S)$.

Given a family $F$, for a subset $S$ of $X$, we write $h(S)$ as the number of subsets $T$ in $X$ such that (1) $T$ differs from $S$ in exactly one element (2) Exactly one set among $S$ and $T$ belongs to $F$.

Test lemma showing that boundaryCount is equivalent to counting subsets $T$ that differ from $S$ in exactly one element and exactly one of $S, T$ belongs to $F$.