Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies

- Assumption A: For all $g\in C_0^1(\Omega)$ we have the product $fg\in L^1(\Omega)$. (Here $C_0^1(\Omega)$ refers to functions which are continuously differentiable in $\Omega$ and extend continuously to $0$ on $\partial\Omega$).

Question 1: Can we show that $f\in L^1(\Omega)$?

Question 2: Does the answer to Question 1 change if we include some or all of the following assumptions:

Assumption B: $f$ possesses a weak derivative which is finite almost everywhere in $\Omega$;

Assumption C: There exists a nonnegative function $f_0 \in H^2(\Omega)\cap C(\bar{\Omega})$ such that $f-f_0=0$ (in the sense of trace) on $\partial\Omega$;

Assumption D: There exists a nonnegative function $h\in H^1(\Omega)$ such that $h$ is nonzero almost everywhere in $\Omega$ and $f=-\ln h$ in $\Omega$.

Note: Assumption D more or less implies Assumption B. I wrote them separately in the hopes of formulating the problem as simply as possible.

Notation: Here $H^k$ is the standard Sobolev space notation for $W^{k,2}$.

9/14/20 Edit:

Question 1 has been answered in the affirmative. I additionally pose the following

Question 3: Answer Questions 1 and 2 in the case that Assumption A is replaced by

- Assumption A': $f\in L^1_{\text{loc}}(\Omega)$.