Since $g<\infty$ a.e the series $$\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})$$ converges a.e in $X$.
From this, why also the series $$f_{n_1}+\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})$$ converges a.e in $X$? can it not happen that at a point $x$ where the series converges we have that $f_{n_1}(x)=\pm \infty$?
My opinion: In a point $x$ in which the series$$\sum_{i=1}^\infty (f_{n_{i+1}}(x)-f_{n_i}(x))<\infty$$converges necessarily every terms of the series must be finite, in particular $f_{n_2}(x)-f_{n_1}(x)$ must be finite and therefore $f_{n_1}(x)$ is finite, so in every point in which the series converges we have that $f_{n_{1}}(x)$ is a number. Therefore, if we denote with $N$ the null set where the series does not converge we have that
$$\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})<\infty\quad\text{in}\quad X\setminus N$$
and as stated before
$$f_{n_1}+\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})<\infty$$ in $X\setminus N$.
This is correct?