Subadditivity of information entropy

For discrete random variables $X,Y$, Shannon entropy satisfiesWhen the two marginal Shannon entropies are finite, the chain rule for information entropy and conditioning reduces entropy prove the inequality, and the difference is the mutual information $I(X;Y)$. Thus equality holds exactly when $X$ and $Y$ are independent random variables.

For example, let $X=Y$ be a fair Bernoulli random variable. Then $H(X,Y)=1$ bit, while $H(X)+H(Y)=2$ bits: the pair does not contain two independent copies of the information. The inequality also holds with extended nonnegative values, but equality of two infinite values does not characterize independence.