It is known that measurable maps preserve independence. That is, if $X_{1}, X_{2}$ are two independent randoms variables, and $f_{1}, f_{2}$ are measurable, then $f_{1} \circ X_{1}$ and $f_{2} \circ X_{2}$ are independent.
But if $X_{1}, X_{2}$ are just uncorrelated, are $f_{1} \circ X_{1}$ and $f_{2} \circ X_{2}$ necessarily uncorrelated? Specifically, can we say $X^{+}_{1}, X^{+}_{2}$, the positive parts, are uncorrelated?
@StubbornAtom gives an example of a pair of uncorrelated dependent r.v.'s https://stats.stackexchange.com/questions/85363/simple-examples-of-uncorrelated-but-not-independent-x-and-y:
$X \in \{-1, 0, 1\}$ discrete and uniformly distributed. $Y = 1$ if $X = 0$ and $Y = 0$ otherwise. Then $X, Y$ are uncorrelated. We can then check $X^{+}, Y^{+}$ are correlated: $\mathbb{E}(X^{+}) \mathbb{E}(Y^{+})$ = $1/9 \neq \mathbb{E}(X^{+}Y^{+}) = 0$.