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Consider the sequence $f_n=(-1)^n \frac{x}{\log(1+x)} \chi_{(0,1/n)}(x)$. Is it true that $$ \sum_n \int_X f_n d\mu= \int_X \sum_n f_n d\mu$$ with $ X=(0,1)$?

I was thinking about using the corollary of the dominated convergence theorem just checking if the $f_n$ are in $L^1$. The integral is quiet a pain though. On my notes it is suggested to solve it considering the sequence of the parcial sums. Any advices?

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Since $\int_X|f_n(x)|\mathrm d\mu\geqslant 1/n$, we cannot use Fubini's theorem. Actually, one can see that the series in the RHS is divergent computing $\sum_{n\geqslant 1}(-1)^n\chi_{(0,1/n)}(x)$ while the serie sin the LHs is convergent (use a summation by parts).

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