Summary: Let \(0 \leq p \leq 1/2 \) and let \(\{0,1\}^n\) be endowed with the product measure \(\mu_p\) defined by \(\mu_p(x)=p^{|x|}(1-p)^{n-|x|}\), where \(|x|=\sum x_i\). Let \(I \subseteq \{0,1\}^n\) be an intersecting family, i.e. for every \(x, y \in I\) there exists a coordinate \(1 \leq i \leq n\) such that \(x_i=y_i=1\). Then \(\mu_p(I) \leq p.\)
Our proof uses measure preserving homomorphisms between graphs.