Are these characteristic functions Lebesgue measurable?

Question

I'm trying to prepare myself for the final exam in real-analysis and I received this problem from my professor, could you help me, please?
Let $E\subseteq[0,1]$ s.t. $E \notin \mathcal{L}(\mathbb{R})$, $A=[-1,1]$ $\setminus$$E$ and let $\chi_A$, $\chi_E$:$\mathbb{R}\to[0,1]$ be the corresponding characteristic functions.
Determine if the functions $\chi_A$, $\chi_E$, $\chi_A$+$\chi_E$, $\chi_A$-$\chi_E$ are Lebesgue measurable.
I was thinking that the characteristic functions are constant, so it follows immediately that are Lebesgue measurable, but I'm not sure if this is correct...