The following question is Exercise 5B from the book Elements of Integration, by R.G. Bartle:
Here is my attempt so far:
We know that $|f|\ge 0$ and so $|f| \in M^+(X,\textbf{X},\mu)$. Also, since $f =0$ $\mu$-almost everywhere on $X$, we also have that $|f|=0$ $\mu$-almost everywhere on $X$.
Since $|f|\in M^+(X,\textbf{X},\mu) $ and $|f|=0$ $\mu$-almost everywhere on $X$., we get from Corollary 4.10 in Bartle (listed below), that $$\int |f| d\mu = 0,$$ i.e. $|f| \in L(X,\textbf{X},\mu)$ and, consequently, $f \in L(X, \textbf{X}, \mu)$.
The only thing left for me to show, is that $$\int f d\mu = 0.$$ Can anyone please help guide me in the right direction?
Corollary 4.10: Suppose that $f \in M^+$. Then $f(x) =0$ $\mu$-almost everywhere on $X$ if and only if $$\int f d\mu = 0.$$
Hint: use the "triangle inequality" for integrals, i.e., $$\left| \int f \right| \leq \int \lvert f \lvert.$$