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.
*Given $f_n$ being continuous on a compact set in $\mathbb{R}$, $f_n$ are Riemann integrable, thus, Lebesgue integrable.
Note we can rewrite $g$ and $h$ as the following:
$g=\limsup f_n=\bigcap_{n\rightarrow\infty}\bigcup_{k\geq n}f_k$, where $k,n\in\mathbb{N}$. Since countable union/intersection of measurable functions are measurable by the definition of $\sigma$-algebra, $g$ therefore is Lebesgue measurable.
Similarly, $h$ is measurable.
Is the above proof correct? Thank you.
Best Answer
First we show that $g_1:=\sup_k f_k$ and $g_2 :=\inf_k f_k$ are measurable. Note that $$g_1^{-1}(a,\infty] = \bigcup_k f_k^{-1}(a,\infty] \quad\text{and}\quad g_2^{-1}[-\infty,a) = \bigcup_k f_k^{-1}[-\infty,a) $$ are measurable. Therefore the countable supremum and infimum of measurable functions are measurable. Consequently $$ \limsup_k f_k = \inf_k(\sup_{j\geq k}f_j) \quad\text{and}\quad \limsup_k f_k = \sup_k(\inf_{j\geq k}f_j) $$ are measurable functions.