Question 10.T.18: For any sequence fn : Ω → ¯R of measurable functions, the fu...
For any sequence f_{n} : Ω → \bar{\mathbb{R}} of measurable functions, the functions\sup f_{n}, \inf f_{n}, \lim \sup f_{n}, \lim \inf f_{n} are all measurable on Ω, and \lim f_{n} is measurable on its domain of existence.
Since f_{n} is measurable for every n ∈ \mathbb{N}, the sets
\left\{x : \sup f_{n}(x) > α\right\} = \overset{∞}{\underset{n=1}{\cup}} \left\{x : f_{n}(x) > α\right\},
\left\{x : \inf f_{n}(x) < α\right\} = \overset{∞}{\underset{n=1}{\cup}} \left\{x : f_{n}(x) < α\right\}
are both measurable, which means that \sup f_{n} and \inf f_{n} are measurable. Noting that
\lim \sup f_{n} = \underset{n∈\mathbb{N}}{\inf} g_{n}, g_{n} = \underset { k≥n}{\sup} f_{k},
\lim \inf f_{n} = \underset {n∈\mathbb{N}}{\sup} h_{n}, h_{n} = \underset{k≥n}{\inf} f_{k},
we conclude that the functions g_{n}, h_{n}, \lim \sup f_{n}, and \lim \inf f_{n} are all measurable. The measurability of \lim \sup f_{n} and \lim \inf f_{n} implies, by Corollary 10.17.1, that
Ω_{0} = \left\{x ∈ Ω : \lim \sup f_{n}(x) = \lim \inf f_{n}(x)\right\},
which is the domain of existence of \lim f_{n}, is measurable. Therefore \lim f_{n} = \lim \sup f_{n}|_{Ω_{0}} is measurable.