Do you have a counterexample for the monotone convergence theorem when you omit the hypothesis that the sequence is increasing? I was thinking about the example where the sequence $f_n$ would approach $f$ as $\frac {\sin(x)} x$ do towards $0$. It appears that the integrals are equal, isn't it?
2026-03-29 10:46:50.1774781210
Counterexample for the monotone convergence theorem
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Take $f_n(x)=\frac{1}{n}\boldsymbol 1_{[0,n]}$. You have that $$\lim_{n\to \infty }f_n(x)=0,$$ but $$\lim_{n\to \infty }\int_{\mathbb R} f_n=1.$$