I don't know how to do the exercise the author gives in the proof. I first thought that if $g$ is a characteristic function over E, then $\int_E \chi_E d\lambda= \lambda(E) = \mu(E) + v(E).$ Similarly, any non-negative measurable function can be approximated by a linear combination of characteristic functions, say $g = \sum_{i=1}^\infty a_i \chi_{E_i}$ for $a_i \in \mathbb{R^+}$ and $\bigcup_{i \in \mathbb{N}} E_i = E$. Is this correct? I appreciate if you give some hint for this.
And, shouldn't the last sentence be $f>0$ on $E \cap X_+$?
Suppose $g$ is a non-negative measurable function on $X$, then there is sequence $\{\phi_n\}_n$ of simple functions such that $0≤\phi_1≤\phi_2≤.......≤g, \phi_n\rightarrow f$ point-wise.
Now we know that $\int_X\phi_n\ d\lambda=\int_X \phi_n\ d\mu+\int_X\phi_n\ d\nu$ for each $n$, since this result holds for indicator function of measurable sets , hence holds also for finite linear combination of indicator functions of measurable sets. Now applying Monotone Convergence Theorem both side we have $$\int_Xg\ d\lambda=lim_{n\rightarrow \infty}\int_X \phi_n\ d\lambda=lim_{n\rightarrow \infty}\int_X \phi_n\ d\mu+lim_{n\rightarrow \infty}\int_X \phi_n\ d\nu\ (since\ both\ limits\ exists)=\int_Xg\ d\mu+\int_Xg\ d\nu$$
You are right , the last statement should be $f>0\ on\ E\cap X_+$