Let $f_n$ be a sequence of real valued measurable functions such that for each $n\in \Bbb N$

$\mu\{x\in X:|f_n(x)-f_{n+1}(x)|>\frac{1}{2^n}\}=\frac{1}{2^n}$.

Show that $f_n$ is pointwise convergent a.e. using Borell-Cantelli Lemma.

I dont understand how to use Borel Cantelli Lemma which states that if $\sum \mu(A_n)<\infty$ then $\mu (\lim_{n\to \infty} \sup A_n)=0$

Can I get some hints ?

  • $\begingroup$ Another way to describe the conclusion about the $\limsup$ of a set is $$ \mu\left(x \in X: \{|f_n(x)-f_{n+1}(x)|>1/2^n\} \mbox{ for infinitely many indices $n$}\right)=0$$ That is $$ \limsup_{n\rightarrow\infty} A_n = \{x \in X : \mbox{$x$ is in infinitely many sets $A_n$}\}$$ $\endgroup$ – Michael May 1 '19 at 5:50

Borel Cantelli Lemma tells you that with probability $1$, $|f_{n+1}(x)-f_n(x)| \leq \frac 1 {2^{n+1}}$ for $n$ sufficiently large Now use the following:

if $(a_n)$ is a sequence of real numbers such that $\sum |a_{n+1}-a_n| <\infty$ then the sequence is convergent.

[ $|a_n-a_{n+m}| \leq |a_n-a_{n+1}|+|a_{n+1}-a_{n+2}|+...+|a_{n+m-1}-a_{n+m}|$. Can you show that this quantity tends to $0$ as $n,m \to \infty$?

More details: let $A_n=\{x:|f_{n+1}(x)-f_n(x)| >\frac 1 {2^{n}}\}$. Then $\sum \mu(A_n) <\infty$. By Borel Cantelli Lemma $\mu (\lim \sup A_n)=0$. Let $E=\lim \sup A_n$. Then $\mu (E)=0$. Suppose $x \notin E$. Using the definition of limsup observe that above basic real analysis lemma can be applied to the sequence $a_n=f_n(x)$. Hence $\lim f_n(x)$ exists whenever $x \notin E$.

  • $\begingroup$ yes because if a series converges then the tail of the series converges to 0 $\endgroup$ – Math_Freak May 1 '19 at 7:51
  • $\begingroup$ Is it correct?sir $\endgroup$ – Math_Freak May 1 '19 at 7:51
  • $\begingroup$ Yes, $\sum \frac 1 {2^{n+1}}$ is convergent so its tails sums tend to $0$. $\endgroup$ – Kavi Rama Murthy May 1 '19 at 7:52
  • $\begingroup$ But how does that solve the original problem,where are we using the Borel lemma,where is the concept of probability used here? $\endgroup$ – Math_Freak May 1 '19 at 7:55

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