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I would like to prove that if $ \delta $ is close enough to zero then Lebesgue Integral: $ \int_{R} {}|f(x) - f(x + \delta)| dx $ is less then $ \epsilon $ for any $ \epsilon>0 $ ,where f belongs to $ L^{1}( R ) $ space

Thank you for all your answers.

reg
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  • What have you tried so far ? Can you prove it for $f = \mathbb{1}_E$ ($E$ measurable of finite measure), or for continuous functions with compact support, for instance ? – Watson Jan 27 '16 at 19:35
  • Yes, I can.If a continuous function has a compact support, then it is uniformly continuous on that set and it goes easy from this moment.And it's easy to prove it for a function, which has a finite value set.But I don't know what to do next. – reg Jan 27 '16 at 19:44
  • Hint : the set $C_c(\mathbb R)$ of continuous functions with compact support is dense in $L^1(\mathbb R)$, meaning that $\forall \epsilon > 0, \exists g \in C_c(\mathbb R) \int |f-g| < \epsilon / 2$. – Watson Jan 27 '16 at 19:53
  • It's not a fact that I should be aware of , but I will definitely try it out.Thank you. – reg Jan 27 '16 at 19:55

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