Does the union of two percolation measures satisfying the (FKG) inequality still satisfy (FKG)?

Question

Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ we have for every $\eta \geq \omega$ that $\eta \in A$. Now a percolation measure $\mu$ satisfies the FKG inequality if for all increasing events $A, B$ we have that $\mu(A \cap B) \geq \mu(A) \mu(B)$. Now, define the union of two percolation measures as the measure with the law coming from sampling each of the two measures independently and taking the union of the 1s.

Q: Does the union of two percolation measures satisfying the (FKG) inequality still satisfy (FKG)?

Answer 1

Welcome to MathOverflow! The answer to your question is yes. Indeed, let $X$ and $Y$ be independent random elements of the set $\{0,1\}^n$ whose distributions are percolation measures, which latter thus satisfy the Harris--FKG inequality, Proposition 2.3: $Ef(X)g(X)\ge Ef(X)\, Eg(X)$ and $Ef(Y)g(Y)\ge Ef(Y)\, Eg(Y)$ for all bounded nondecreasing functions $f$ and $g$. We have to show that for any such functions $f$ and $g$ we have $$Ef(X\vee Y)g(X\vee Y)\ge Ef(X\vee Y)\, Eg(X\vee Y),\tag{1}$$ where $X\vee Y$ is coordinate-wise maximum of $X$ and $Y$. Note that (i) your union of the measures is the distribution of $X\vee Y$ and (ii) the inequality $P(X\vee Y\in A\cap B)\ge P(X\vee Y\in A)P(X\vee Y\in B)$ for increasing sets $A$ and $B$ is a special case of (1), with $f$ and $g$ being the indicator functions of $A$ and $B$.

To prove (1), let $F(y):=Ef(X\vee y)$ and $G(y):=Eg(X\vee y)$ for $y\in\{0,1\}^n$. Then \begin{align} Ef(X\vee Y)g(X\vee Y)&=\sum_{y\in\{0,1\}^n} P(Y=y)Ef(X\vee y)g(X\vee y) \\ &\ge\sum_{y\in\{0,1\}^n} P(Y=y)Ef(X\vee y)\,Eg(X\vee y) \\ &=\sum_{y\in\{0,1\}^n} P(Y=y)F(y)\,G(y) \\ &=EF(Y)G(Y) \\ &\ge EF(Y)\,EG(Y) \\ &= Ef(X\vee Y)\, Eg(X\vee Y), \end{align} so that (1) is proved. The first inequality in the above multiline display follows by the Harris--FKG inequality for $X$ and nondecreasing functions $x\mapsto f(x\vee y)$ and $x\mapsto g(x\vee y)$, and the second inequality there is an instance of the Harris--FKG inequality for $Y$ and nondecreasing functions $F$ and $G$.