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Let $\{f_n\}$ be a sequence of measurable functions with $$\int_{R^d} |f_n| \leq c_n \quad \text{with} \ \ c_n \downarrow 0$$

As we know this doesn't imply $f_n \rightarrow 0$ a.e. (when $d=1$ consider $\chi_{[0,1]}, \chi_{[0,\frac{1}{2}]}, \chi_{[\frac{1}{2},1]}, \chi_{[0,\frac{1}{4}]}$...)

What additional conditions on $\{c_n\}$ would guarantee almost everywhere convergence?

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I believe summability is enough. If $\sum_n c_n $ converges, then $$ \sum \int |f_n| = \int \sum |f_n| \leq \sum c_n < \infty. $$ The first equality is by the monotone convergence theorem. Then, as the middle integral is finite, we must have $\sum_n |f_n| <\infty$ a.e., which means that $f_n \to 0 $ a.e.

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  • $\begingroup$ Thank you, this is good enough. $\endgroup$ – user16015 Oct 11 '15 at 5:39
  • $\begingroup$ How do you apply the monotone convergence theorem? To apply that, should you already assume the pointwise convergence of $\sum |f_n|$ which is what is to be proved? $\endgroup$ – Hans Jul 23 '17 at 22:18
  • $\begingroup$ I believe we don't need convergence in the usual sense, since you can have functions taking values in $[0,\infty]$. $\endgroup$ – Andrew Jul 24 '17 at 17:19

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