An upper bound for chi-square divergence in terms of KL divergence for general alphabets In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ defined over a general alphabet $\mathcal{X}$, if $P\ll Q$, then 
$$\chi ^2(P||Q):=\int_{\mathcal{X}}\Big(\frac{dP}{dQ}\Big)^2dQ$$ and 
$$D(P||Q):=\int_{\mathcal{X}}dP\log\frac{dP}{dQ}.$$

I am looking for an upper bound of $\chi^2(P||Q)$ in terms of $D(P||Q)$ which works wven if $\mathcal{X}$ is uncountable. 
What I need is a special case where $P=P_{XY}$ and $Q=P_X\times P_Y$, for two random variables with joint and product distributions are $P_{XY}$ and $P_X\times P_Y$, respectively. Noticing that in this case KL divergence is equal to the mutual information , I need an upper bound of chi-square divergence in terms of mutual information.
Your definition of $\chi^2$ divergence is missing a term; namely,
$$
\chi^2(P\|Q) = \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ} - 1\right)^2
= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
$$
(see e.g. this Wikipedia article on $f$-divergences).

With this in hand, recall that by concavity of the logarithm, we have
$$
\log x \leq x-1, \qquad \forall x &gt;0
$$
(where $\log$ denotes the natural logarithm), and thus
$$\begin{align}
D(P\| Q) &amp;=  \int_{\mathcal{X}} dP\log\frac{dP}{dQ}\\
&amp;\leq  \int_{\mathcal{X}} dP\left(\frac{dP}{dQ} - 1\right) \\
&amp;= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
\end{align}$$
showing that the inequality holds:
$$
D(P\| Q) \leq \chi^2(P\|Q)\,,\qquad \forall P\ll Q
$$