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Here's the problem.

Let $(X, \mathscr{A}, \mu)$ be a measure space, and let $f,f_n$ be nonnegative functions that belong to $\mathscr{L}^1(X, \mathscr{A}, \mu, \mathbb{R})$. Prove that if $f_n \to f$ $\mu$-almost everywhere and $\int f_n \to \int f$, then $\int |f_n - f| \to 0$.

Note that $|f_n - f| \leq f_n + f$, so $g_n := f_n + f - |f_n -f| \geq 0$. Fatou gives $\int \liminf_n g_n \leq \liminf_n \int g_n$. Notice that $g_n \to 2f$, almost everywhere. Since the limit inferior is sub-additive, this means: $$ \int 2f = \int \liminf_n g_n \leq \liminf_n \int g_n \leq \int 2f + \liminf_n - \int |f_n - f| $$ Since $\liminf -a_n = -\limsup a_n$, we have $\limsup_n \int |f_n -f| \leq 0$, which implies the claim.

It took me a while to find this solution. Assuming it is correct, can someone help me understand why the definition of $g_n$ is natural in some sense here? I think this proof mechanically works, but isn't very intuitive to me.

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  • $\begingroup$ How did you get $\int 2f$ on the RHS of the last inequality? $\endgroup$ – Bungo Aug 28 '18 at 20:52
  • $\begingroup$ @Bungo, Well $\int g_n = \int f_n + \int f - \int |f_n - f|$, right? The assumption is that $\int f_n \to \int f$, agreed? So $\liminf \int g_n \leq \lim \int f_n + \int f + \liminf- \int |f_n -f| = \int 2f + \liminf - \int |f_n -f|$, agreed? $\endgroup$ – Drew Brady Aug 28 '18 at 21:29
  • $\begingroup$ Ah right, looks fine. $\endgroup$ – Bungo Aug 28 '18 at 21:31
  • $\begingroup$ The proof is correct. But yeah, pretty unintuitive, you're right. $\endgroup$ – amsmath Aug 28 '18 at 22:25
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It's easy to verify that $$\min(x,y) = {x+y-|x-y|\over2}$$ so $$g_n=2\min(f,f_n)$$

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  • $\begingroup$ Good point. I wonder if we can make the proof go through directly with $\min(f, f_n)$. $\endgroup$ – Drew Brady Aug 28 '18 at 23:07

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