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Let $\mu$ be a measure function. I want to show that and $\mu(\lim_{n\rightarrow\infty}\sup A_n)\geq \lim_{n\rightarrow\infty}\sup\mu(A_n)$ provided that $\mu(\cup^\infty_{n=1}A_n)<\infty$.

I have already proved that $\mu(\lim_{n\rightarrow\infty}\inf A_n)\leq \lim_{n\rightarrow\infty}\inf\mu(A_n)$ from answers given here. Can I use this to prove that $\mu(\lim_{n\rightarrow\infty}\sup A_n)\geq \lim_{n\rightarrow\infty}\sup\mu(A_n)$?

Could you please give me a hint on how to start proving this.

user3911153
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    This is reverse Fatou's lemma specialized to characteristic functions. Reverse Fatou's lemma says that if $(f_n)$ is a sequence of nonnegative measurable functions and $|f_n| \leq g \in L^1(X, \mu)$, then $\int \limsup f_n \geq \limsup \int f_n$. It is easy to prove using Fatou's lemma. – Mason Oct 30 '21 at 04:21
  • @Mason, why is the condition $\mu(\cup^\infty_{n=1}A_n)<\infty$ provided in the question? – user3911153 Oct 30 '21 at 04:42
  • That condition gives you your $g \in L^1(X, \mu)$. – Mason Oct 30 '21 at 04:44

1 Answers1

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Let $B=\bigcup_n A_n$. Then $B \setminus \lim \sup A_n=\lim\inf (B\setminus A_n)$. So $$\mu( B \setminus \lim \sup A_n)$$ $$ \leq \lim \inf \mu( B \setminus A_n)$$ $$=\lim \inf (\mu (B)-\mu (A_n))=\mu (B)-\lim \sup \mu(A_n).$$

Can you finish? [$\mu (B) <\infty$ is needed at this last step]]

Kavi Rama Murthy
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  • So now we have $\mu( B \setminus \lim \sup A_n) \leq \mu (B)-\lim \sup \mu(A_n)$ then $\mu( B) -\mu(\lim \sup A_n) \leq \mu (B)-\lim \sup \mu(A_n)$ hence $\mu(\lim \sup A_n) \geq \lim \sup \mu(A_n)$ is this right? – user3911153 Oct 30 '21 at 04:54
  • @user3911153 Yes, that is exactly the argument. – Kavi Rama Murthy Oct 30 '21 at 04:55
  • but where did we use $\mu(\cup A_n)<\infty$. Is it so we can subtract $\mu(\cup A_n)$ from both sides? – user3911153 Oct 30 '21 at 04:57
  • @user3911153 Yes, $ \infty -1=\infty -2$ but $1 \neq 2$, right? You cannot subtract infinity from both sides of an equation. – Kavi Rama Murthy Oct 30 '21 at 04:59
  • Just one last question. If it is not too much trouble for you could you please explain how we get $B \setminus \lim \sup A_n=\lim\inf (B\setminus A_n)$ and $\mu( B \setminus \lim \sup A_n) \leq \lim \inf \mu( B \setminus A_n)$. – user3911153 Oct 30 '21 at 05:05
  • For the first one verify that each side is contained in the other (or use De Morgan's Laws). The second one is exactly the inequality you stated in line 3 of your question with $B \setminus A_n$ in place of $A_n$. – Kavi Rama Murthy Oct 30 '21 at 05:09