Lemma 4.2 (Shearer, expectation version). Assuming that:

$X = (X_{1}, \dots, X_{n})$ a random variable
$A \subset [n]$ a randomly chosen subset of $[n]$ , according to some probability distribution (don’t need any independence conditions!)
for each $i \in [n]$ , $ℙ [i \in A] \geq μ$

Then

H [X] \leq μ^{- 1} 𝔼_{A} H [X_{A}] .