So I have stumbled upon the following theorem:
Let $\left\{f_n\right\}$ be a sequence of measurable functions. For $x \in X$, put
$$
g(x) = \sup \left\{ f_n (x) \mid n \in \mathbb{N} \right\} \\
h(x) = \limsup_{n \to \infty} f_n (x)
$$
Then $g$ and $h$ are measurable
The proof is as follows:
$$
\left\{ x \mid g(x) > a \right\} = \bigcup_{n=1}^{\infty} \left\{ x \mid f_n (x) > a \right\}
$$
At least to me, this seems to not be true, so I was wondering if someone could help me understand where I go awry.
It seems to me that you could have a sequence of functions $f_n (x) = x + \frac{x}{n} : n \neq 3 , f_3 (x) = 1000$. From the proof
$$
g(x) = \sup \left\{ x + \frac{x}{n} \mid n \in \mathbb{N} \right\} = \left\{ \begin{array}{rr} 2x & : x \ge 0 \\ x & : x < 0 \end{array} \right.
$$
Now it seems to me that each $f_n (x)$ is measurable in $\mathbb{R}$ however the statement in the proof isn't true here. Put $a = 3$ then $\left\{ x \mid g(x) > 3 \right\} = \left( \frac{3}{2}, \infty \right)$ however $\bigcup_{n=1}^{\infty} \left\{ x \mid f_n (x) > 3 \right\} = (- \infty , \infty )$ because $f_3 (x) > 3 \; \forall \; x \in \mathbb{R}$, so wouldn't this be a counter example to the proof given?
Thanks in advance!
Best Answer
You have a sequence $\langle f_n\rangle$ and you set $$g(x)=\sup_{n\geqslant 1}\{f_n(x)\}$$
The claim is that
First, suppose $x$ is such that $g(x)>a$. This means $$\sup_{n\geqslant 1}\{f_n(x)\}>a$$
This means that there exists $m$ such that $f_m(x)>a^{\dagger}$, so $x\in \{x\mid f_m(x)>a\}$ and thus $x$ is in that union. Conversely, suppose $x$ is in the union. Then for some $m$ we have $f_m(x)>a$, thus $\sup\limits_{n\geqslant 1 }\{f_n(x)\}=g(x)>a$.
$\dagger$: Note that if for each $n$ we had $f_n(x)\leq a$ then the $\sup$ would be $\leq a$. Thus, by contraposition, we must have $f_m(x)>a$ for at least one $m$.