Let us have two random variables $X$ and $Y$ and their joint probability density function $f_{XY}(x,y)$. Let us assume that the function $f_{XY}$ can be written as a product of two functions $g(x)$ and $h(y)$. Then we say that the random variables $X$ and $Y$ are independent. But what if the range of $y$ in the domain of $f_{XY}$ is dependent of $x$? For example $f_{XY}$ is defined on the set $\{|x|\leq 1, |y| \leq x^2\}$, are $X$ and $Y$ still independent?
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You would write your joint density $f_{XY}$ as \begin{align*} f_{XY}(x,y) = g(x)h(y) \textbf{1}_{\{|x|\leq 1, |y| \leq x^2\}}, \end{align*} which can no longer be split up into a product of functions depending only on one variable (since you can't factor the indicator into two indicators).
