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let $X$ be a finite measure space and $\{f_n\}$ be a sequence of integrable functions, $f_n \rightarrow f\text{ a.e.}$ on $ X$. I want to show if (1) holds, then (2) holds too.

$$\lim_{n \rightarrow \infty}\int_X |f_n| \, d\mu=\int_X |f| \, d\mu,\tag{1}$$

$$\lim_{n \rightarrow \infty}\int_X |f_n-f| \, d\mu=0.\tag{2}$$

My attempt:

I have proven that (2) holds for nonnegative $f$. Then for the general case, I split the set to $E^+=\{x: f \geq 0\}$ and $E^-=\{x: f \leq 0\}$:

$$\lim_{n \rightarrow \infty}\int_{E^+} f_n \, d\mu-\int_{E^+} f \, d\mu -\lim_{n \rightarrow \infty}\int_{E^-} f_n \, d\mu+\int_{E^-} f \, d\mu=0$$

But I don't know how to proceed from here!

  • 0
    Your sets $E^+$ and $E^-$ depend on $n$. Is it intended?2012-10-31
  • 1
    @did: sorry, it was a typo.2012-10-31
  • 0
    You should also assume that $f$ is integrable, otherwise the result won't be true.2012-10-31
  • 0
    Also, if you already have it for non-negative $f$, for the general case you could consider the sequence $(f_n + |f|)$2012-10-31
  • 1
    This is possibly a duplicate of http://math.stackexchange.com/q/51502 and http://math.stackexchange.com/q/2220392012-11-01

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