Working through a problem in Terry Tao's Measure theory book and I've reduced one of the exercises (Borel Cantelli Lemma) to proving $\lim \limits_{n \to \infty} \mu (\bigcup_{i=n}^\infty E_i) = 0$. By the countable subadditivity property of measures this amounts to showing that $\lim \limits_{n \to \infty} \sum_{i=n}^\infty \mu (E_i) = 0$. We are given as an assumption that $\sum_{i=0}^\infty \mu (E_i) < \infty$. Hence the question.
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Consider $x_i=\frac{1}{2^i}$ – Blue Cat Blues May 27 '23 at 10:25
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Yes, thank you Martin. – Nicholas Yager May 27 '23 at 10:34
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A. it is only true if the series is absolutely convergent. Since measures are non-negative it applies here.
B. hint: think of partial sums as Cauchy sequences.
ryaron
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2Statement (A) is incorrect. All convergent series have the property. – Sassatelli Giulio May 27 '23 at 11:18
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$\sum x_i<\infty$ is not synonymous with convergence of $\sum x_i$. – Kavi Rama Murthy May 27 '23 at 11:26
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It's true that $\sum x_i < \infty$ does not imply convergence in general, but the specific case I'm working with $x_i$ are strictly positive. – Nicholas Yager May 27 '23 at 11:35
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What if the partial sums oscillate (and is bounded) but never converges ? – Lelouch May 27 '23 at 11:35
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@geetha290krm Ah, right, I was not considering the possibility $\sum_{i=1}^\infty x_i=-\infty$, although my statement is still true as is. – Sassatelli Giulio May 27 '23 at 20:29
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@SassatelliGiulio $\sum x_i$ may also be undefined (neither $+\infty$ nor $-\infty$). You should always say in words that $\sum x_i$ is convergent. – Kavi Rama Murthy May 27 '23 at 23:12
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@geetha290krm I don't agree that $\sum_{i=1}^\infty x_i<\infty$ should mean $\limsup_{n\to\infty} \sum_{i=1}^n x_i<\infty$ any more than I could agree that $\lim_{x\to\infty} f(x)<\infty$ should mean $\limsup_{x\to\infty} f(x)<\infty$. If you speak of an object, that object should be there. – Sassatelli Giulio May 27 '23 at 23:28
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Bounded series are not necessarily convergent. Take for example the sequence ${-(1)^n\frac{1}{2}}$ it's series is bounded but can have 3 possible values. – ryaron May 28 '23 at 08:18