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

Question

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?

Answer 1

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).