The Shannon entropy of a random variable $X$ has much behaviour analogous to a signed measure. Previous work has concretized this connection by defining a signed measure $\mu$ on an abstract information space $\tilde{X}$, which is taken to represent the information that $X$ contains. This construction is sufficient to derive many measure-theoretical counterparts to information quantities such as the mutual information $I(X; Y) = \mu(\tilde{X} \cap \tilde{Y})$, the joint entropy $H(X,Y) = \mu(\tilde{X} \cup \tilde{Y})$, and the conditional entropy $H(X|Y) = \mu(\tilde{X}\, \setminus \, \tilde{Y})$. We demonstrate that there exists a much finer decomposition with intuitive properties which we call the logarithmic decomposition (LD). We show that this signed measure space has the useful property that its logarithmic atoms are easily characterised with negative or positive entropy, while also being coherent with Yeung's $I$-measure. We present the usability of our approach by re-examining the G\'acs-K\"orner common information from this new geometric perspective and characterising it in terms of our logarithmic atoms. We then highlight that our geometric refinement can account for an entire class of information quantities, which we call logarithmically decomposable quantities.

翻译：随机变量$X$的香农熵具有许多类似有符号测度的性质。先前的研究通过在抽象信息空间$\tilde{X}$上定义有符号测度$\mu$（该空间代表$X$所包含的信息）具体化了这一联系。这一构造足以推导出许多信息量的测度论对应形式，例如互信息$I(X; Y) = \mu(\tilde{X} \cap \tilde{Y})$、联合熵$H(X,Y) = \mu(\tilde{X} \cup \tilde{Y})$以及条件熵$H(X|Y) = \mu(\tilde{X}\, \setminus \, \tilde{Y})$。我们证明存在一种具有直观性质的更精细分解，称之为对数分解(LD)。研究表明，该有符号测度空间具有一个有用性质：其对数原子的负熵或正熵易于刻画，同时与Yeung的$I$测度保持相容。通过从这一新几何视角重新审视Gács-Körner公共信息，并依据我们的对数原子对其进行刻画，我们展示了该方法的实用性。最后我们指出，这种几何精细化可以解释一整类信息量，我们称之为对数可分解量。