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

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)?

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 $$G$$.