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Let $\{f_i\}$ be a sequence of pointwise discontinuous functions whose limit is Dirichlet's function. I read that

$$\lim_{n\to\infty}\int f_n(x)dx \not= \int \lim_{n\to\infty} f_n(x)dx$$

as the right hand side attempts to integrate Dirichlet's function, which is not (Riemann) integrable. I get that part.

I don't really understand why the left hand makes sense though. I see that for any finite $i$, $f_i$ it is discontinuous on a non-dense set, so it's integrable, but when $i$ goes to infinity it seems to me like this integral shouldn't exist.

I guess one confusion is it seems like $\frac{d}{dx}(\lim_{n\to\infty}\int f_n(x)dx)$ should equal $\lim_{n\to\infty}f_n(x)$ which I suppose can't be true, but it doesn't seem obvious why.

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    Take a look at the [Dominated Convergence Theorem](http://en.wikipedia.org/wiki/Dominated_convergence_theorem)2012-10-05
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    @Patrick: Not sure I understand. I get that they may be Lebesgue integrable; I just don't understand why it's not Riemann integrable.2012-10-05

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