Uncertainty Function satisfies Axioms of Uncertainty

Theorem
Let $X$ be a random variable.

Let $X$ take a finite number of values with probabilities $p_1, p_2, \dotsc, p_n$.

Let $\map H X$ be the uncertainty function of $X$:


 * $\map H X = \ds -\sum_k p_k \lg p_k$

where:
 * $\lg$ denotes logarithm base $2$
 * the summation is over those $k$ where $p_k > 0$.

Then the uncertainty function satisfies the Axioms of Uncertainty.