Entropy monotonicity under independent addition

If $U,V$ are independent random variables taking values in a countable additive group, conditioning on $V$ makes $U+V$ a translate of $U$. Consequently $H(U+V\mid V)=H(U)$, and conditional entropy gives $H(U+V)\ge H(U)$. If the two information entropies are finite, equality holds exactly when $U+V$ and $V$ are independent.

// Target: probability-and-statistics.bigb