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Let $\varphi$ be a simple function with canonical form $\varphi(x)=\sum_{k=1}^na_k\chi_{E_k}$.

The triangle inequality claims that $|\int\varphi|\leq\int|\varphi|$.

In the proof of this statement given in Stein and Shakarchi, the authors claim that $|\varphi|=\sum_{k=1}^N|a_k|\chi_{E_k}(x)$, but I do not see why this is true. Isn't it only true if the $a_k$ are all nonnegative?

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  • $\begingroup$ Remember we can always take the sets over which the simple functions are defined to be pairwise disjoint. $\endgroup$ Commented Jun 23, 2019 at 17:06

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$|\varphi|(x)=|\varphi(x)|=|\sum_{k=1}^na_k\chi_{E_k}(x)|$. Suppose $x\in E_j, 1\le j\le n$. Then $|\varphi|(x)=|a_j|=|a_j|\chi_{E_j}(x)=\sum_{k=1}^n|a_k|\chi_{E_k}(x).$

Since this holds for any arbitrary $x $, we have $|\varphi |=\sum_{k=1}^n|a_k|\chi_{E_k}$.

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