Biased measure of a set family (${\mu }_p(\mathcal{F})={\sum }_{A\in \mathcal{F}}{p}^{|A|}(1-p{)}^{n-|A|}$)

The biased measure of a set family is its probability when each ground-set element is independently included with probability $p$. If $a_j$ is the proportion of the $j$th level occupied by the family, then ${\mu }_p(\mathcal{F})={\sum }_j\left(\genfrac{}{}{0ex}{}{n}{j}\right)a_j{p}^j(1-p{)}^{n-j}$. This translates a combinatorial level bound into a probability inequality for independent Bernoulli random variables.