i have a question about demostration of the properties of step functions, if i have two functions with they standar representation $\phi=\sum a_iX_{A_i}$ and $\psi=\sum b_jX_{A_j}$ and i do a partition taking the union of finite intervals of $\phi$ and $\psi$, such that $\phi=\sum c_rX_{A_{c_r}}$ and $\psi=\sum d_rX_{A_{c_r}}$, the product is:
$\psi \phi=\sum c_rX_{A_{c_r}} \sum d_rX_{A_{c_r}}....(*)$ How can i argue that (*) is the same that $\psi \phi=\sum c_rd_rX_{A_{c_r}} $ because the product of sums are not equals. Thank you.
In general, $X_{A}X_{B} = X_{A\cap B}$. In the case of partitions, the only non-zero products are given when $A=B$.