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?
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- 6$\begingroup$ Pointwise limits of measurable functions are measurable. Apply that to the partial sums. $\endgroup$Daniel Fischer– Daniel Fischer2013-08-01 15:59:51 +00:00Commented Aug 1, 2013 at 15:59
- $\begingroup$ Functions from where to where? And by sum of functions, do you mean like $(f+g)(x) = f(x) + g(x)$ or direct sum of functions on the direct sum of the domains? $\endgroup$Daniel Robert-Nicoud– Daniel Robert-Nicoud2013-08-01 16:00:11 +00:00Commented Aug 1, 2013 at 16:00
- $\begingroup$ I am not sure if your first sentence is so straight forward; I would prefer to have the math for this and know about what type of function etc. you are talking? $\endgroup$al-Hwarizmi– al-Hwarizmi2013-08-01 16:02:27 +00:00Commented Aug 1, 2013 at 16:02
- $\begingroup$ Functions from an arbitrary set to the extended real-domain and I mean $(f+g)(x)=f(x)+g(x)$. $\endgroup$Simon– Simon2013-08-01 16:03:25 +00:00Commented Aug 1, 2013 at 16:03
- $\begingroup$ You need some assumptions here, otherwise your infinite sum will not converge, think of $f_n(x)=1$. See en.wikipedia.org/wiki/Dominated_convergence_theorem for details. $\endgroup$Moishe Kohan– Moishe Kohan2013-08-01 16:13:24 +00:00Commented Aug 1, 2013 at 16:13
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1 Answer
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1 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.
- 1$\begingroup$ It should be fine for the partial sums to converge to a value in the extended reals i.e. $\infty$ or $-\infty$, is that right? In which case the pointwise limit is the constant function $x \mapsto \infty$. $\endgroup$jII– jII2018-10-22 00:04:16 +00:00Commented Oct 22, 2018 at 0:04