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This is an attempt to go further than this answer.

Essentially, we have either a summation of an integral:

$$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$

...or an integral of a summation:

$$\int{ \left( \sum_x{ f(x) } \right) dx } \tag{2}$$

The answer, without going into detail, states that we can generally interchange the summations and integrations if $f(x)$ is positive. I'm wondering if it's ever possible to interchange summation and integration if $f(x)$ happens to be negative for some values, and what these cases are. I've shown each case separately as (1) and (2), in case that one direction is different than the other.

Community
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Matt Groff
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1 Answers1

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Yes, just look at the answer just above the one you gave the link to : if $\int \sum |f_n| < \infty$ or $\sum \int |f_n| < \infty$ (which are two equivalent conditions according to Tonelli's theorem as stated in the answer), then you can interchange summation and integral.

In usual settings (domain of $f$ is in a complete measured space, $f$ is valuated in a euclidian space, etc...), you can derive it from either using Dominated Convergence Theorem, or Fubini/Tonelli theorems where one of the measured space is $\mathbb{N}$ with the counting measure. AFAIK, there is no other well-known and general theorem with such simple hypothesis that ensures interchangeability.

yago
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  • I looked at that answer and missed that - thank you! I'm wondering what types of books cover this - would this be covered in real analysis, for instance? I'm wondering what all may cover this in detail... – Matt Groff Jun 02 '14 at 11:29
  • You're welcome. It was part of my analysis course when I was undergraduate, and I remember that we used quite often (altogether with other theorems derivated from TCD) so I guess it should be in any decent (real) analysis book, yes. – yago Jun 02 '14 at 11:43
  • Books about measure theory usually contain this stuff. Also the Wiki http://en.wikipedia.org/wiki/Fubini_theorem#Fubini.27s_theorem_for_integrable_functions may be useful. – Urgje Jun 02 '14 at 11:44