Measure convergence and pointwise convergence in almost everywhere

Asked Jun 16 '20 at 18:37

Active Jun 16 '20 at 18:37

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im in a trouble. I want to prove a proposition about Lebesgue integral, it would be great if the following were true:

I have the definition of measure convergence of a functions sequence on E:

$ \forall e>0\forall d>0 $ exist N such that: $ n≥N \implies m(${$x\in E:|f_{n}(x)-f(x)|≥e$})<d

The doubt that I have is Can I say $lim_{n\rightarrow \infty} f_{n}=f$ in almost all points?

In other words If A:={x$\in E:(lim_{n\rightarrow \infty} f_{n})(x)≠f(x)$} Then m(A)=0?

Thanks <2+1

asked Jun 16 '20 at 18:37

dedekind1

This is not true. See here: https://math.stackexchange.com/questions/1415891/almost-everywhere-convergence-versus-convergence-in-measure – PhoemueX Jun 16 '20 at 18:41
Also this answer formulated in probabilistic terms – Michh Jun 16 '20 at 18:43

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