$$\int^a_0{\left(\int^b_0{f(y)\mathrm{d}y}\right)g(x)\mathrm{d}x}$$
In this relationship, $f(y)$ and $g(x)$ are probability distribution functions (PDF). If $x$ and $y$ are 'dependent' random variables, can we still say that:
$$\int^a_0{\left(\int^b_0{f(y)\mathrm{d}y}\right)g(x)\mathrm{d}x}=\left(\int^a_0{g(x)\mathrm{d}x}\right)\left(\int^b_0{f(y)\mathrm{d}y}\right)$$