How is the sum of two Lebesgue integrable functions?

Question

I'm practicing for the final exam in real-analysis and I am at the chapter Measure Theory and Integration. I found this exercise, but I don't know how to solve it...Could you please help me?
Let $(X, \cal{A}, \mu)$ be a measure space and $f, g : X → \mathbb{R}$. Determine if the following implications hold in general:
(i) both functions $f$ and $g$ are integrable $⇒ f + g$ is integrable;
(ii) $f + g$ is integrable $⇒$ at least one of the functions $f$ or $g$ is integrable;

Answer 1

I assume that when you say integrable you mean that the integral of the function is finite.

I belive that for the item $(i)$ you can use the property that the integral is additive and you will get your answer.

For $(ii)$ I belive it is false because you can consider $f=-x$ and $g=x$ and you get that $f+g=0$.