KL Divergence with different domains

I want to calculate KL Divergence between a normal and an exponential r.v. i.e. $$D(P||Q) = ?\\ \;\; P=N(\mu,\sigma), \;\; Q=exp(\lambda)$$ My problem is that in this case the domains of the distributions are different - the domain of $$P$$ is $$x\in R$$ and the domain of Q is $$x \in [0,\infty )$$. Which domain should I integrate over? If this is the domain of $$P$$ the value of $$\log(Q(x)/P(x))$$ is not defined.

Let's say we use want to calculate the KL Divergence for $$\mu = 1, \sigma = 2 ,\lambda =1$$ what will be the result? I can calculate $$D(Q||P)$$ but it is not the same.

In this case the KL-divergence $$D(P||Q)$$ is indeed infinity. To have a well defined (not infinity) KL-divergence, we need support$$(P)\subseteq$$support$$(Q)$$. Here "support" means whereever the probability is non-zero. See discussion here (page 3).
• Note that this implies that if we calculated $D(Q||P)$ we would not have a problem, although how meaningful the result would be is open to question! – jbowman Sep 29 '18 at 18:45