Solution
= Solution
\b[No such <sequence> exists.] By <Fatou's lemma> on $(0,1)$,
$$
\int_0^1\liminf_n|g_n(x)|\,dx
\leq\liminf_n\int_0^1|g_n(x)|\,dx
\leq\liminf_n\|g_n\|_1=0.
$$
But the proposed nonnegative measurable integrand is positive at every point of $(0,1)$, so its <integral> cannot be zero. Explicitly, the sets where it is at least $1/k$ cover $(0,1)$; at least one has positive measure, forcing a positive <integral>. This contradicts the displayed bound.