I have difficulty interpreting the following question:
If the simple function $\phi$ in $M^{+}(X,\mathcal{A})$ has the (not necessarily standard) representation $\phi=\displaystyle\sum_{k=1}^{n}b_k\chi_{F_k}$ where $b_k \in \mathbb{R}$ and $F_k \in \mathcal{A}$, show that
$\int \phi \mathrm{d}\mu = \displaystyle\sum_{k=1}^{n}b_k\mu({F_k}).$
We know that a simple function $\phi$ is of $\bf{standard \;\;representation}$ if
$\phi=\displaystyle\sum_{k=1}^{n}a_k\chi_{E_k}$ where $a_k$ are distinct and $E_j$ are disjoint nonempty subsets of $X$. Moreover, its integral is of the form $\int \phi \mathrm{d}\mu = \displaystyle\sum_{k=1}^{n}a_k\mu({E_k}).$
Is the question stating that the integral will hold for $\phi$ where $b_k$ are not distinct and where $F_k$ are not disjoint? I would like to say that this will hold when $b_k$ are not distinct, but I do not think this will hold if $F_k$ are not disjoint sets.