Question:
Proposition 2.13 - Let $\phi$ and $\psi$ be simple functions in $L^+$.
a.) If $c\geq 0$, $\int c\phi = c\int \phi$.
b.) $\int(\phi + \psi) = \int \phi + \int \psi$.
c.) If $\phi\leq \psi$, then $\int \phi\leq \int \psi$.
d.) The map $A\rightarrow \int_{A}d\mu$ is a measure on $M$.
Attempted proof a.) - If $c\geq 0$ then $$\int c\phi = \int c\sum_{j}a_j \chi_{E_j} = c\int \sum_{j}a_j \chi_{E_j} = c\int \phi$$
Attempted proof b.) Let $\phi = \int_{j}a_j\chi_{E_j}$ and $\psi = \sum_{k}b_k\chi_{f_k}$, then $$\int \phi + \int \psi = \int \sum_{j}a_j\chi_{E_j} + \int \sum_{k}b_k\chi_{F_k} = \sum_{j}a_j\mu(E_j) + \sum_{j}b_k\mu(F_k)$$
Before I go on with c and d I saw that after this last step we can say $$\sum_{j}a_j\mu(E_j) + \sum_{j}b_k\mu(F_k) = \sum_{j}a_k\sum_{k}\mu(E_j\cap F_k) + \sum_{k}b_k\sum_{j}\mu(E_j\cap F_k)$$I don't understand how they are equal. So this is a more of an algebra question I guess more than a measure theory question. Any suggestions is greatly appreciated.
Background Information:
We fix a measure space $(X,M,\mu)$, and we define $$L^+ = \ \ \text{the space of all measurable functions from} \ X \ \text{to} \ [0,\infty]$$If $\phi$ is a simple function in $L^+$ with standard representation $\phi = \sum_{1}^{n}a_j\chi_{E_j}$, we define the integral of $\phi$ with respect to $\mu$ by $$\int \phi d\mu = \sum_{1}^{n}a_j\mu(E_j)$$