Prove that if \[ f,g:\mathcal P(n)\longrightarrow\mathbb R \] are real-valued functions on the power set of \([n]\), then \[ g(A)=\sum_{B\subseteq A}f(B) \qquad\text{for every }A\subseteq[n] \] if and only if \[ f(A)=\sum_{B\subseteq A}(-1)^{|A\setminus B|}g(B) \qquad\text{for every }A\subseteq[n]. \]
A subcube of \(\mathcal P(n)\) is a subset \[ \mathcal C_{A,B}=\{C\subseteq[n]:A\subseteq C\subseteq B\} \] for some \(A\subseteq B\subseteq[n]\). It is trivial if \(A=B\), and hence \(|\mathcal C|=1\).
Characterize all functions \[ h:\mathcal P(n)\longrightarrow\mathbb R \] such that for every nontrivial subcube \(\mathcal C\) of \(\mathcal P(n)\), \[ \sum_{C\in\mathcal C}h(C)=0. \]