Edited:
Given that $f(x_1,x_2,...)=f_1(x_1)f_2(x_2)...$ and $g(x_1,x_2,...)=g_1(x_1)g_2(x_2)...$ are always positive.
Also, $f$ and $g$ are continuous everywhere.
If
$[\displaystyle\prod_{n=1}^∞\int_{-1}^{1}dx_n] f(x_1,x_2,...) =∞$ and
$[\displaystyle\prod_{n=1}^∞\int_{-1}^{1}dx_n] g(x_1,x_2,...) =∞$,
is it necessarily true that
$[\displaystyle\prod_{n=1}^∞\int_{-1}^{1}dx_n] f(x_1,x_2,...)g(x_1,x_2,...) =∞$ ?
For more tricky case, if $g$ is not always positive or negative and
[\displaystyle\prod_{n=1}^∞\int_{-1}^{1}dx_n] g(x_1,x_2,...) =±∞ (oscillating),
is it true that
[\displaystyle\prod_{n=1}^∞\int_{-1}^{1}dx_n] f(x_1,x_2,...)g(x_1,x_2,...) =±∞?