Alternate definition of Entropy

The obvious reason $-\sum_{i,\,j}P_{i,\,j}\ln P_{i,\,j}$ is a more common definition is that it's extensive, but we often don't care about that, so much as about relative entropies. In that case, one may as well add a constant $c$ to taste, giving $c-\sum_{i,\,j}P_{i,\,j}\ln P_{i,\,j}=-\sum_{i,\,j}P_{i,\,j}(\ln P_{i,\,j}-c)$. The choice $c=1$ has the unique attraction that the whole expression is just $-\sum_{i,\,j}\int_0^{P_{i,\,j}}\ln pdp$.