I recently came across the following question: Find a sequence of integrable functions $(f_n)_n$ such that $$\int_a^b|f_n(x)|dx\to0\quad\text{but}\quad f_n\not\to0\quad\text{a.e.}$$ I am unsure how to approach this question, and would really appreciate a hint.
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Do you know the standard example of a sequence which converges in measure but not almost everywhere? – Kavi Rama Murthy Oct 22 '22 at 11:50
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1Have $f_n(x) = 1$ on a smaller and smaller subinterval, but make sure the subinterval walks around the whole $[a, b]$ interval. – Tomek Czajka Oct 22 '22 at 11:51
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3A quintessential example is the typewriter sequence. – Sangchul Lee Oct 22 '22 at 11:52
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1If you are not insistent on the space $(\Bbb{R},\mathcal{B},\lambda)$ , then for Probability spaces, there are easier(to understand) examples . Consider a sequence of independent $\text{Bernoulli}(\frac{1}{n})$ random variables. Then their means(integrals wrt Probability measure) converge to $0$, but they don't converge almost surely due to second Borel Cantelli lemma. In any case, just talking about existence of such random variables require the notion of Product measures and the Daniell-Kolmogorov existence theorem. – Mr. Gandalf Sauron Oct 22 '22 at 12:23
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4Take a look at Does convergence in $L^p$ imply convergence almost everywhere?. Also see the typewriter sequence. – Sarvesh Ravichandran Iyer Oct 22 '22 at 12:23
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Hint: Try the following sequence of functions $[0,1]\to\mathbb{R}$:
$f_1=\chi_{[0,\frac{1}{2}]}, \ f_2=\chi_{[\frac{1}{2},1]}, \ f_3=\chi_{[0,\frac{1}{3}]}, \ f_4=\chi_{[\frac{1}{3},\frac{2}{3}]}, \ f_5=\chi_{[\frac{2}{3},1]}, \ f_6=\chi_{[0,\frac{1}{4}]}, \ f_7=\chi_{[\frac{1}{4}, \frac{2}{4}]}, ...$
Where $\chi$ is the characteristic function of a set.
Mark
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