Sum of two quasi-integrable functions

Let $f,g: X\to \overline{\mathbb{R}}=\mathbb{R}\cup \{\infty, -\infty\}$ be quasi-integrable functions w.r.t. the measure space $(X,\mathcal{M},m)$, i.e., $\int_X f= \int_X f^+-\int_X f^-$ and $\int_X g=\int_X g^+-\int_X g^-$ are both well-defined (do not have the form $\infty-\infty$).

I want to show that $f+g$ is quasi-integrable and $\int_X (f+g)=\int_X f+\int_X g$, if both $f+g$ and $\int_X f+\int_X g$ are also well-defined (do not have the form $\infty-\infty$).

Here is my attempt:

(1) It is easy to show that $\int_X f+\int_X g$ is well-defined iff $\left(\int_X f^+<\infty\land \int_X g^+<\infty\right)$ or $\left(\int_X f^-<\infty\land \int_X g^-<\infty\right)$.

(2) It is also easy to show that $(f+g)^+\leq f^++g^+$ and $(f+g)^-\leq f^-+g^-$. Thus, $\int_X (f+g)^+\leq \int_X f^++\int_X g^+$ and $\int_X (f+g)^-\leq \int_X f^-+\int_X g^-$. Combined with (1), it follows that $f+g$ is quasi-integrable.

(3) W.L.O.G., assume $\int_X f^+<\infty\land \int_X g^+<\infty$. If $\int_X f^-<\infty$ and $\int_X g^-<\infty$, then $f,g\in L^1$, which degenerates into the trivial case. So it suffices to show that the result remains correct when at least one of $\int_X f^-$ and $\int_X g^-$ is $\infty$.

(4) W.L.O.G, assume $\int_X f^-=\infty$. Thus, $\int_X f+\int_X g=-\infty+\int_X g=-\infty$. We need to show that $\int_X (f+g)=-\infty$, which is equivalent to $\int_X (f+g)^-=\infty$.

However, I am stuck at this point. In a word, how could I show that $\int_X (f+g)^-=\infty$ when $\int_X f^+<\infty$, $\int_X g^+<\infty$, and $\int_X f^-=\infty$.

Or is there a better way to solve this problem?

Thanks in advance!