Can you help me please?
If $f \in M^{+}(X, \mathbf{X})$ and $$\int f d\mu \lt +\infty,$$ then the set $N=\{x \in X: f(x)\gt 0\}$ is $\sigma$-finite (that is, there exists a sequence $(F_n)$ in $\mathbf{X}$ such that $N\subset \cup F_n$ and $\mu (F_n) \lt +\infty$).
Hint: Let $F_n = \{x:f(x) \geqslant 1/n\}.$
Can you show that $F_n$ is measurable and $N \subset \cup F_n?$
Then justify and use
$$\frac1{n} \mu(F_n) \leq \int_{F_n}f d\mu\leq\int_{X}f d\mu.$$