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Strengthened the simple function approximations.

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Bokuan Li
2026-06-28 14:33:32 -04:00
parent 4acc8fdf31
commit 1d740724b4
4 changed files with 37 additions and 31 deletions
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src/measure/bochner-integral/strongly.tex
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@@ -24,7 +24,7 @@
By the separable case, $f$ is $(\cm, \cb_{F})$-measurable. Let $B \in \cb_E$, then $B \cap F \in \cb_F$ by \autoref{lemma:borel-induced}. Therefore $\bracs{f \in B} = \bracs{f \in B \cap F} \in \cm$, and $f$ is $(\cm, \cb_E)$-measurable.
(2) $\Rightarrow$ (3): By \autoref{proposition:measurable-simple-separable-norm}.
(2) $\Rightarrow$ (3): By \autoref{corollary:measurable-simple-separable-norm}.
(3) $\Rightarrow$ (1): For each $\phi \in E^*$, $\phi \circ f = \limv{n}\phi \circ f_n$ is measurable by \autoref{proposition:limit-measurable}. Since
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