The KL divergence is not a metric because e.g. it does not satisfy the symmetry property that metrics posses. According to the definition of measure, the KL divergence doesn't seem to be a measure, although the related Wikipedia article introduces the KL divergence as "a measure of how one probability distribution is different from a second". So, mathematically, what is the KL divergence?
Note that I know how the KL divergence is defined, so don't tell me how it's defined.