Let $(X,A,\mu)$ a measure space. Assume a sequence $(f_n)$ of positive (i.e. $\geq 0$) measurable functions from $X$ to $\mathbb{R}$ converges to $f$ in $L^p$. Then is $f$ positive ?
My idea is the following : we know that a subsequence $(f_{n_k})_k$ converges pointwise almost everywhere to $f$, but each $f_{n_k}$ is also positive, so $f$ is positive almost everywhere. Is that true ?
We have $$f_n\to f,\ f_n=|f_n|\to|f|$$ The second convergence follows from the triangle inequality $|\,|f_n|-|f|\,|\le |f_n-f|$. Hence $f=|f|\ge 0.$
I assume that by positive you mean $\geq0$ (the other possible meaning leads trivially to a negative answer).
Now, consider the measurable subset $E=f^{-1}(]-\infty,0[)$.
Then for all $n$,
$\lvert\lvert f-f_n\rvert\rvert_p\geq(\int_X 1_E\lvert f-f_n\rvert^p d\mu)^\frac{1}{p}\geq(\int_X 1_E\lvert f\rvert^p d\mu)^\frac{1}{p}>0$
as $\mu(E)>0$ and $1_E\lvert f\rvert^p>0$ on $E$.
Absurd.
So $f$ is positive in the sense of $L^p$ function (positive up to a measure $0$ subset).
