(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For a sequence ##\{f_n\}## of measurable functions with common domain ##E##, show that the following functions are measurable: ##\inf \{f_n\}##, ##\sup \{f_n\}##, ##\lim \inf \{f_n\}##, and ##\lim \sup \{f_n\}##

2. Relevant equations

3. The attempt at a solution

It suffices to show that ##\sup \{f_n\}## and ##\lim \sup \{f_n\}## are measurable, since the negative of a measurable function is measurable and ##\inf \{f_n\} = - \sup \{-f_n\}## and ##\lim \inf \{f_n\} = - \lim \sup \{-f_n\}##. First we show that ##h(x) := \sup \{f_k(x) \mid k \in \Bbb{N} \}## is measurable. Define the function ##g_n(x) = \max \{f_1(x),...,f_n(x) \}## which is measurable for every ##n##. First note that ##\{f_1(x),...,f_n(x) \} \subseteq \{f_k(x) \mid k \in \Bbb{N} \}## and therefore ##\max \{f_1(x),...,f_n(x) \} \le \sup \{f_k(x) \mid k \in \Bbb{N} \}## or ##h(x) -g_n(x) \ge 0## for every ##n \in \Bbb{N}##. Let ##x \in E## and ##\epsilon > 0## be arbitrary. Then there exists an ##N \in \Bbb{N}## such that ##h(x) < f_N(x) + \epsilon##. And if ##n \ge N##, then ##g_n(x) \ge f_N(x)## or ##g_n(x) + \epsilon \ge f_N(x) + \epsilon > h(x)## or ##|h(x) - g_n(x)| < \epsilon##. This proves that ##g_n## converges pointwise to ##h##, which means that ##h## is measurable.

To see that ##\lim \sup \{f_n\}## is a measurable function, recall that for each it is defined as ##\lim_{n \infty} \sup \{f_k(x) \mid k \ge n \}## which is by definition the pointwise limit of the sequence ##(\sup \{f_k(x) \mid k \ge n \})_{n \in \Bbb{N}}## of measurable functions.

Does this seem right? I solved the problem and then did a google search to find a solution. I found a couple, but proofs were slightly different from what I came up with, so I just wanted to have my solution verified.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Sup. and Lim. Sup. are Measurable Functions

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**