Chi-square $\chi^2$ divergence dual formulation

Question

For two probability measures $P$ and $Q$, the chi-square divergence is given by $$\chi^2(Q,P) = \int \left( \frac{\mathrm{d}Q}{\mathrm{d}P}−1\right)^2 \mathrm{d}P.$$ I heard there is a dual formulation which, I vaguely remember, looks like $$\chi^2(Q,P) =\sup_{g\in L^2(Q)}\frac{(\mathbb{E}_P g- \mathbb{E}_Q g)^2}{\mathbb{E}_Pg}$$ where $\mathbb{E}_P$ and $\mathbb{E}_Q$ are expectations w.r.t. $P$ and $Q$, respectively.

I am looking for a reference for this dual formulation.

Answer 1

This is not correct. The denominator sholud be $\sqrt {E_P g^{2}}$ and the supremeum is over $g \in L^{2}(P), g \neq 0$. The proof is a simple application of Cauchy -Schwarz inequality.