(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.

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# Sup. and Lim. Sup. are Measurable Functions

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