combine the integrands

Question

If the integral looks like this: $$ \iint \{\iint f(x)f(y)\mathrm{d}x\mathrm{d}y\}g(x)g(y) \mathrm{d}x\mathrm{d}y $$ is it legitimate to 'combine' the integrals as $$ \iint f(x)f(y)g(x)g(y) \mathrm{d}x \mathrm{d}y ? $$

The assumption is that both $f$ and $g$ are non-negative, and are elements of $L^2([0,1])$ space ( i.e. they are square-integrable).

Answer 1

Of course not. Just take $[0,1]$ and $f=0$ on $[0,1/2]$ , $f=1$ on $[1/2,1]$. Now take $g=1-f$. Then $\int\int f(x)f(y)g(x)g(y)dxdy=0$, whereas the quadruple integral is not zero.

Answer 2

No, you cannot do this. Suppose

$$ f(x) = \begin{cases} 1 &:0 \leq x \leq \dfrac12 \\ 0 &: \dfrac12 < x \leq 1 \end{cases}$$

and $g(x) = 1 - f(x)$. Then the first integral is $\dfrac{1}{16}$, the second is zero.