# Is the countable sum of a set of measurable functions also measurable?

It is rather straightforward to show that the sum of two measurable functions is also measurable. Therefore we can extend the logic to say that $\sum\limits_{i=1}^n f_i$ is measurable providing $f_i$ is measurable. However can we take $n\rightarrow\infty$ and say that $\sum\limits_{i=1}^\infty f_i$ is a measurable function?

#### Solutions Collecting From Web of "Is the countable sum of a set of measurable functions also measurable?"

As Daniel pointed out, the pointwise limit of measurable functions is measurable.

If we define $g_n=\sum_{i=1}^n f_i$, then as long as $\sum_{i=1}^{\infty} f_i = \lim_{n\to\infty}g_n$ converges, it defines a measurable function.

You need some assumptions here, otherwise your infinite sum will not converge, think of $f_n(x)=1$. See https://en.wikipedia.org/wiki/Dominated_convergence_theorem for details.