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Let $(\Omega, \mathcal{F})$ be a measurable space and let $A_i,B_j$ be in $\mathcal{F}$ such that $\Omega = \uplus_{i=1}^n A_i = \uplus_{j=1}^m B_j$, ie they're finite disjoint unions. Let $s$ be a simple function and $s = \sum_{i=1}^n \alpha_i 1_{A_i}$ and $s = \sum_{j=1}^m \beta_j 1_{B_j}$, be two partitions of $s$, where $1_A(x) = 1, x\in A; =0, x\notin A$. Then s also has the partition $s = \sum_{i,j}^{n,m} \alpha_i 1_{A_i \cap B_j}$, and a similar one for the $\beta_j$. But doesn't this mean $\alpha_i = \beta_j$ for all $i,j$ and so this simple function is not interesting? To see this let $\omega \in A_i \cap B_j$, then $s(\omega) = \alpha_i 1_{A_i \cap B_j}(w) = \alpha_i = $ similar for $\beta_j$. But I'm sure I got something wrong.

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It means that $\alpha_i=\beta_j$ for all pairs $(i,j)$ such that $A_i\cap B_j$ is non-empty.

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  • $\begingroup$ Thanks. I got scared for a moment. $\endgroup$ Commented Jul 21, 2013 at 1:31
  • $\begingroup$ No problem, glad to help. $\endgroup$ Commented Jul 21, 2013 at 1:35

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