# Prove that if $(f_n)_{n=0}^\infty$ is a uniformly bounded sequence of measurable functions, then $f=\limsup f_n$ is measurable.

What's the difference between the following two questions:

Let $f_n:[0,1]\rightarrow\mathbb{R}$ be a sequence of continuous functions. Prove that $g=\limsup f_n$ and $h=\liminf f_n$ are Lebesgue measurable.

Prove that $g=\limsup f_n$ and $h=\liminf f_n$ are Lebesgue measurable.

Prove that if $(f_n)_{n=0}^\infty$ is a uniformly bounded sequence of measurable functions, then $f=\limsup f_n$ is measurable.

*My question is why do I need "uniformly bounded" to prove $\limsup f_n$ is measurable?

• I suspect it is to avoid having $f$ take infinite values. This assumption isn't needed if we are okay with $f$ being defined into the extended real line $[-\infty,\infty]$. Aug 9, 2017 at 20:49
• @JohnGriffin Thank you! Aug 9, 2017 at 20:51

Just to make sure that $\limsup_{n\in\mathbb N}f_n$ is a real function (in the sense that it only takes real values).