Let $A$ has positive measure, $(A_n)_n$ is a pairwise disjoint sequence subset of $A$ of positive measure. Let $S, T\subset \mathbb{N}$ and $S\neq T$, and $\chi_S, \chi_T$ is the characteristic function of $\cup_{n\in S}A_n$. Compute $||\chi_S-\chi_T||_\infty$ where $||.||_\infty$ is $L^\infty$-norm.
I guess that $$||\chi_S-\chi_T||_\infty=1$$ by replace $S,T$ by some explicit sets. Please help me.
The norm $L^\infty$ norm of a characteristic function $\chi_A$ is either $1$ or $0$, depending on if the measure of of $A$ is positive or null. Consider what $\chi_T - \chi_S$ is, in terms of a single characteristic function. You are indeed right in your guess.