Can a Subsequence of Measurable Functions Converge in L1?

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Homework Statement



Let fn be a sequence of measurable functions converges to f a.e. Is it possible to get a subsequence fnk of fn s.t. fn converges in L1 ?

2. The attempt at a solution
I have proved the converse statement is true and i guess the above statement is impossible but I fail to prove it.
 
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Hint: Look for a counterexample where ##f_n \rightarrow 0## pointwise but ##\|f_n\| = 1## for all ##n##.
 
Thanks for your suggestion
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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