$\chi _M + \chi _N$ iff $M$ and $N$ are measurable Let $(\mathbb {R}^d, \mathcal {M}, \mu )$ be a measure space with $\mathcal {E} \subset \mathcal {M}$. $M$ and $N$ are disjoint subset in $\mathbb {R}^d$.
$\chi _M + \chi _N$ and $\chi _M + 2 \chi _N$ should be measurable iff $M$ and $N$ are measurable right?
($\mathcal {E}$ is the set of all unions of finite intervals.)

Yes.  If $\chi_M + \chi_N$ and $\chi_M + 2\chi_N$ are both measurable, then $(\chi_M + 2\chi_N) - (\chi_M + \chi_N) = \chi_N$ is also measurable, which then implies $\chi_M$ is measurable as well.  So $M$ and $N$ are measurable.
