[1]https://i.stack.imgur.com/nW1sW.png [proof of 1.40][1] I got stuck in the last ≤. And another: how can $E=f^{-1}(Δ)$ dosen't contradict with $E∈\mathfrak{M}$?
2026-04-19 17:05:20.1776618320
Theorem 1.40 from Rudin RCA, the last ≤
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Because for $x\in E=f^{-1}(\Delta)$, then $|f(x)-\alpha|\leq r$ by the definition of $\Delta$, so $\dfrac{1}{\mu(E)}\displaystyle\int_{E}|f-\alpha|d\mu\leq\dfrac{1}{\mu(E)}\cdot r\cdot\mu(E)=r$.