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Suppose we are on a finite measure space and $f_n\to f$ in measure. I want to show there is a subsequence such that $f_{n_k}\to f$ a.e.

I know that $f_n\to f$ in measure $\iff$ $\int_X\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu\to0$. So if $f_n\to f$ in measure, then by Fatou's lemma, $\int_X\text{lim inf}\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu\leq\text{lim inf}\int_X\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu=0$. Since $\text{lim inf}\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}\geq 0$, this means $\text{lim inf}\frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}=0$. Can I infer from this that there is a subsequence such that $\frac{|f_{n_k}(x)-f(x)|}{1+|f_{n_k}(x)-f(x)|}\to 0$ and so $f_{n_k}\to 0$ almost everywhere?

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No, your argument does not give a convergent subsequence. (You can only get a subsequence depending on x). Here is a valid proof: Choose $n_k$ such that $\mu \{|f_n -f| >2^{-k}\} < 2^{-k}$ for $n \geq n_k$. Since $\sum_k \mu \{|f_{n_k} -f| >2^{-k}\} < \infty$ it follows the set of points x for which $|f_{n{_k}}(x)-f(x)| > 2^{-k}$ has measure 0 which shows that $f_{n{_k}} -f \to 0$ a.e..

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Is your function $f$ non-negative? Without that, you can not apply Fatou's lemma. The answer to your question is yes, I think you may find this post helpful Do we need to have a subsequence such that $\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|$?

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  • $\begingroup$ I'm applyling Fatou's lemma to $g_n=\frac{|f_n-f|}{1+|f_n-f|}$, which is a non-negative sequence, so $\text{lim inf} g_n\geq 0$ $\endgroup$ – Tom Chalmer Feb 26 '18 at 1:31
  • $\begingroup$ I know it's true for sequences, but I don't see why it's necessarily true for pointwise convergence. It seems like the subsequence $g_{n_k}$, and hence, $f_{n_k}$, would depend on the point $x$ you're evaluating at $\endgroup$ – Tom Chalmer Feb 26 '18 at 1:33

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