If I have a sequence $\{E_n\}_{n=1}^\infty$ of $\mathcal{M}$-measurable sets, and $E=\cup_{n=1}^\infty E_n$. How can I show that $\chi_{E_n} f\rightarrow\chi_Ef$ pointwise?
2026-04-05 10:28:35.1775384915
I have a question about point wise convergence concerning Lebesgue measurable sets.
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Suppose that the $E_n$ are increasing (otherwise it is false).
If $x\in E$, then $\chi_{E_n}(x) = 0$ as long as $x\notin E_n$, and then $\chi_{E_n}(x) = 1$.
This happens for every $x\in E=\cup E_n$.
If $x\notin E$, then $x\notin E_n$ and the sequence stays at 0.
In every case, $$\chi_{E_n}(x)f(x)\to \chi_{E}(x)f(x) $$