Let $(f_n)_{n=1}^\infty$ be a sequence of measurable, real-valued functions. Prove that each of the following sets is measurable:
1) $A = \{x: f_n(x) \to \infty \text{ as } n \to \infty\}$;
2) $B = \{x: f_n(x) \text{ is eventually irrational}\}$;
3) $C = \{x: f_n(x) > 0 \text{ for infinitely many } n\}$.
what I have tried:
I think the solution is to write each of the above as a countable intersection/union of some sets of the form $\{x: f_n(x) < \text{ or }> \text{ or } \leq \text{ or } \geq a\}$ for some $a \in \mathbb R$.
For $A$, $f_n(x) \to \infty$ if and only if there exists an increasing real sequence $a_n \to \infty$ and $f_n(x) \geq a_n$ for all $n$. Then I attempted to use this observation to write $A$ as an intersection of $\{x: f_n(x) \geq a_n\}$ over $n$ and then take the union over all such sequences. But the latter union is clearly not countable so I'm stuck.
For $B$, every irrational number is the limit of a sequence of rationals, then $\{x: f_n(x) = a_n\} = \{x: f_n(x) \geq a_n\} \cap \{x: f_n(x) \leq a_n\}$ and I think the rest should use some arguments similar to those in 1) so I'm also stuck.
For $C$, $C = \bigcup_{n=1}^\infty \bigcap_{m = n}^\infty \{x: f_m(x) > 0\}$, which is clearly measurable since each $f_m$ is and the union and intersection are both countable.
Could you please provide some hints for $A$, $B$ and also tell me if my reasoning for $C$ is correct? Thank you so much.
Best Answer
Hints:
The function $\mathbb{R} \to \overline{\mathbb{R}}$ given by $x\mapsto \liminf_{n\to\infty} f_n(x)$ is measurable so
$$A = \{x : \liminf_{n\to\infty} f_n(x) = +\infty\}$$
is measurable.
$(f_n(x))_n$ is eventually irrational if and only if $(f_n(x))_n$ is irrational for all except finitely many $n \in \mathbb{N}$. Therefore $$B = \liminf_{n\to\infty} \{x : f_n(x) \in \mathbb{R}\setminus\mathbb{Q}\} = \bigcup_{n=1}^\infty \bigcap_{k=n}^\infty \{x : f_k(x) \in \mathbb{R}\setminus\mathbb{Q}\}$$
is measurable.
Similarly
$$C = \limsup_{n\to\infty} \{x : f_n(x) > 0\} = \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty \{x : f_k(x) > 0\}$$
is measurable.