This paper proves that increasing functions on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property. Applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model are given. The paper begins with an introduction that discusses recent inequalities for the correlations of Ising ferromagnets and their generalizations. It also mentions Harris' inequality for percolation models. The main result is the generalization of the positivity of correlations to the case where the set is a finite distributive lattice and the measure satisfies a convexity condition. This result generalizes previous findings for the random cluster model. Section 2 recalls lattice-theoretic notions and proves the main result, showing that the sufficient condition for the positivity of correlations is not necessary. Section 3 discusses applications, including Ising spin systems and percolation and random-cluster models. The paper defines a partially ordered set as a lattice if any two elements have a least upper bound and a greatest lower bound. A sublattice is a subset that is itself a lattice. A semi-ideal is a subset where if an element is in the subset and another element is less than or equal to it, then the other element is also in the subset. A distributive lattice satisfies certain conditions involving the operations ∧ and ∨. A real function on a partially ordered set is increasing if for any x ≤ y, f(x) ≤ f(y), and decreasing if f(x) ≥ f(y).This paper proves that increasing functions on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property. Applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model are given. The paper begins with an introduction that discusses recent inequalities for the correlations of Ising ferromagnets and their generalizations. It also mentions Harris' inequality for percolation models. The main result is the generalization of the positivity of correlations to the case where the set is a finite distributive lattice and the measure satisfies a convexity condition. This result generalizes previous findings for the random cluster model. Section 2 recalls lattice-theoretic notions and proves the main result, showing that the sufficient condition for the positivity of correlations is not necessary. Section 3 discusses applications, including Ising spin systems and percolation and random-cluster models. The paper defines a partially ordered set as a lattice if any two elements have a least upper bound and a greatest lower bound. A sublattice is a subset that is itself a lattice. A semi-ideal is a subset where if an element is in the subset and another element is less than or equal to it, then the other element is also in the subset. A distributive lattice satisfies certain conditions involving the operations ∧ and ∨. A real function on a partially ordered set is increasing if for any x ≤ y, f(x) ≤ f(y), and decreasing if f(x) ≥ f(y).