Under what condition can converge in $L^1$ imply converge a.e.?

Suppose that $$\int_{R^d} \sum_{n=1}^\infty |f_n(x)|\,dx= \sum_{n=1}^\infty \int_{R^d} |f_n(x)|\,dx \leq \sum_{n=1}^\infty c_n<\infty.$$

This shows that $\sum_{n=1}^\infty |f_n(x)|<\infty$ for almost every $x\in R^d$, and hence $|f_n(x)|\to0$ for such $x$.