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In this book http://books.google.com.br/books/about/Fundamentals_of_Applied_Functional_Analy.html?id=Od5BxTEN0VsC&redir_esc=y

I found the following theorem (page 48):

Theorem(Partial converse of Lebesgue theorem): Assume that $(u_k)$ is a sequence in $L^{p}(\Omega)$ ($\Omega$ open subset of $R^n$ , $1 \leq p < \infty$) converging to a function $u$ (convergence in $L^p$). Then there exists a subsequence converging almost everywhere on $\Omega$, which is dominated by an $L^p$ function.

In this theorem is considered the Lebesgue measure.

This theorem is valid in $R^n $for a arbitrary measure ? if yes , somenone can say to me a book? I tried to search , but i dont found anything ...

my english is terrible , sorry ..

math student
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    It's general. You pick a subsequence such that $\sum \lVert f_{n_k} - f_{n_{k+1}}\rVert < \infty$, then $g = \lvert f_{n_0}\rvert + \sum \lvert f_{n_k} - f_{n_{k+1}}\rvert \in L^p$ and that dominates the sequence. – Daniel Fischer Aug 07 '13 at 22:03

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As Daniel pointed out it is more general than assuming the Lebesgue measure. It's a standard result and you can find it for instance in the book of Haim Brezis, 'Functional Analysis, Sobolev Spaces and Partial Differential Equations'. Check out Theorem 4.9.

Serban
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