2
$\begingroup$

Here is the version of reverse Fatou's lemma I am looking at.

$E_n$ is a sequence of events. $P(\limsup E_n) \geq \limsup P(E_n)$

Here is Fatou's lemma.

Let $f_1,f_2,\ldots$ be a sequence of non-negative measurable functions on a measure space $(S, \Sigma, \mu)$. Define the function $f:S \to [0,1]$ a.e. pointwise limit by :$$f(s) =\liminf_{n\to\infty} f_n(s),\qquad s\in S.$$ Then $f$ is measurable and $$\int_S f\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu\,.$$

What is the connection of the two theorems?

$\endgroup$
  • $\begingroup$ I believe you can define $f_n(s) = 1\{s \in E_n^c\}$, where $1\{\}$ denotes an indicator function that is either 1 or 0 depending on the event inside the braces. $\endgroup$ – Michael Jun 20 '16 at 23:14
  • $\begingroup$ Can you write out what is in your mind? I do not see the reverse relationship in here. $\endgroup$ – user1559897 Jun 20 '16 at 23:16
  • $\begingroup$ I mean apply Fatou to those functions and see what happens. For example, can you compute $\int f_n(s) d\mu$ ? $\endgroup$ – Michael Jun 20 '16 at 23:16
  • $\begingroup$ By the same reasoning you can prove $P[\liminf E_n] \leq \liminf P[E_n]$, which is a more direct application of Fatou as it does not need complements. To build intuition, you might want to prove that one first. $\endgroup$ – Michael Jun 20 '16 at 23:27
  • $\begingroup$ I know the proof of inverse fatou's lemma already. And ∫fn(s)dμ is 1 - P($E_n$) if I get you correctly. $\endgroup$ – user1559897 Jun 20 '16 at 23:31
0
$\begingroup$

To each event $E_n$ there is an "indicator" random variable $$ X_n = \begin{cases} 1 & \text{if the event $E_n$ occurs,} \\ 0 & \text{if the event $E_n$ does not occur.} \end{cases} $$ Let $Y_n=1-X_n$. Then $\liminf\limits_{n\to\infty} Y_n \vphantom{\dfrac 1\int}$ is a random variable that has some probability of being equal to $1$ and is otherwise $0$. And $\operatorname{E}(Y_n) = 1 - \Pr(E_n) = \Pr(\text{not } E_n)$. Since a random variable is a function whose domain is a probability space and an expected value is its integral over that space, the version of Fatou's lemma for measurable functions says $$ \operatorname{E} \left(\liminf_{n\to\infty} Y_n\right) \le \liminf_{n\to\infty} \operatorname{E} (Y_n). $$ Now show that

  • the event $\limsup\limits_{n\to\infty} E_n$ occurs if and only if the random variable $\liminf\limits_{n\to\infty} Y_n$ is equal to $0$; and
  • $\operatorname{E}\left( \liminf\limits_{n\to\infty} Y_n \right) = \Pr\left(\liminf\limits_{n\to\infty} (\text{not }E_n) \right) = \Pr\left( \text{not } \left(\limsup\limits_{n\to\infty} E_n\right) \right)$. Those last named events have equal probabilities because they are the same event: either one of them occurse if and only if the other one occurs.

In other words, the version of Fatou's lemma for measurable functions quickly entails the version for events.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.