Convergence of the sum implies convergence of the summands?

Question

Given a sequence $\{g_{n}\}$ of Lebesgue measurable functions such that $\sum_{n=1}^{\infty}\int_{I}|g_{n}|$ is convergent, then does that mean $\sum_{n=1}^{\infty}\int_{I}g_{n}^{+}$ and $\sum_{n=1}^{\infty}\int_{I}g^{-}_{n}$ are also convergent? Here $f^{+}=\max(f,0)$ and $f^{-}=\max(-f,0)$.

I am trying to prove something which will be done if I have got this point right.

Answer 1

The answer is yes since $$ g_{n}^{-}\leq |g_n|,\qquad g_{n}^{+}\leq |g_n|. $$