# $L^1$ convergence and almost everywhere convergence

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?

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.
• 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? – Hans Jul 23 '17 at 22:18
• I believe we don't need convergence in the usual sense, since you can have functions taking values in $[0,\infty]$. – Andrew Jul 24 '17 at 17:19