Definitions of K-L divergence based on likelihood ratio and on R-N derivative From Wikipedia

For distributions $P$ and $Q$ of a continuous random variable, KL-divergence is defined to be the integral: 
  $$
    D_{\mathrm{KL}}(P\|Q) = \int_{-\infty}^\infty \ln\left(\frac{p(x)}{q(x)}\right) p(x) \, {\rm d}x, \!
$$
  where $p$ and $q$ denote the densities of $P$ and $Q$.
More generally, if $P$ and $Q$ are probability measures over a set $X$, and $P$ is absolutely continuous with respect to $Q$, then the Kullback–Leibler divergence from $P$ to $Q$ is defined as
  $$
    D_{\mathrm{KL}}(P\|Q) = \int_X \ln\left(\frac{{\rm d}P}{{\rm d}Q}\right) \,{\rm d}P, \!
$$
  where $\frac{{\rm d}P}{{\rm d}Q}$ is the Radon–Nikodym derivative of $P$ with respect to $Q$, and provided the expression on the right-hand side exists.

If I understand correctly, R-N derivative is same as likelihood ratio, but the latter can still exist when the former doesn't exist, i.e. when one measure isn't absolute continuous wrt the other measure.
The first definition of K-L divergence is based on likelihood ratio, and the second definition is based on R-N derivative. So how is the second "more general"?
Thanks and regards!
 A: The first definition only applies to probability measures on $\mathbb{R}$ which have densities. Not all probability measures are defined on $\mathbb{R}$, and not all probability measures defined on $\mathbb{R}$ have densities. 
A: My guess is that Wikipedia failed to mention some important conditions on their definition for the continuous random variable case. I believe the required condition for the first definition to be well-defined is
\begin{align}
P&\ll Q\ll \lambda,
\end{align}
where $\lambda$ is Lebesgue measure (and perhaps by abuse of notatation the random variables are substituted for the measures they induce). If this were not true, then $q(x)$ could be 0 on a set of nonzero Lebesgue measure for which $p(x)$ is not 0. Then the integrand would be $\infty$ on a set with larger than 0 measure. So the first definition fails to be well-defined (at least as a real valued function) without the added condition. 
So, to answer the question, the second definition is more general because it only requires the condition $P\ll Q$, and not the condition $Q\ll\lambda$.
