Condition for Independence from Product of Expectations/Corollary/Converse

Converse of Corollary to Condition for Independence from Product of Expectations
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$ such that:
 * $E \left({X Y}\right) = E \left({X}\right) E \left({Y}\right)$

Then it is not necessarily the case that $X$ and $Y$ are independent.

Proof
Proof by Counterexample:

Let $X$ be a discrete random variable whose distribution is defined as:
 * $p_X \left({-1}\right) = p_X \left({0}\right) = p_X \left({1}\right) = \frac 1 3$

Let $Y$ be the discrete random variable defined as:
 * $Y = \begin{cases}

0 & : X = 0 \\ 1 & : X \ne 0 \end{cases}$

We have:

So $X$ and $Y$ are dependent.

But:

So $E \left({X Y}\right) = E \left({X}\right) E \left({Y}\right)$ but $X$ and $Y$ are not independent.