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**slevvio** Let $\displaystyle S = \displaystyle\bigcup_{r=1}^n E_n$ be the union of distinct intervals $\displaystyle E_1, \ldots E_n $ in $\displaystyle \mathbb{R}$ i.e. none of the intervals are the same

I am trying to show that $\displaystyle \chi_S $ is a step function $\displaystyle \implies E_1,\ldots E_n$ are disjoint. I am a bit stumped.

So assume $\displaystyle \chi_S = \lambda_1 \chi_{E_1} + \ldots + \lambda_n \chi_{E_n}$.

If there is only one non-empty intersection of two intervals, it is easy to show that this is impossible, but I am unable to prove it in the general case, i.e. when there is an arbitrary number of collections of intervals with non-empty intersection, has anyone got any tips? Thanks very much