P (X \in A \cap B) \geq P (X \in A) P (X \in B) .

Here is an induction proof of the product-measure version, also called Harris' inequality. The result is immediate for one coordinate. For

n

coordinates, define

a_{j} = P (A ∣ X_{n} = j), b_{j} = P (B ∣ X_{n} = j), j \in {0, 1} .

n - 1

coordinates, so the induction hypothesis gives

P (A \cap B) \geq (1 - p_{n}) a_{0} b_{0} + p_{n} a_{1} b_{1} .

Monotonicity gives

a_{1} \geq a_{0}

and

b_{1} \geq b_{0}

. The difference between the right side and

P (A) P (B) = ((1 - p_{n}) a_{0} + p_{n} a_{1}) ((1 - p_{n}) b_{0} + p_{n} b_{1})

p_{n} (1 - p_{n}) (a_{1} - a_{0}) (b_{1} - b_{0}) \geq 0.

This closes the induction and proves positive association.

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