Prove that the following sets are measurable

Question

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.

Answer 1

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.

Answer 2

In case of $A$, I give you a hint. Note that $f_n(x)\rightarrow \infty$ as $n\rightarrow \infty$ if and only if for all $k\in \mathbb{N}$ there is an $m\in \mathbb{N}$ such that for all $p>m$ we have $f_p (x)>k$.

In case of $B$, I again give you a hint: there are countably many rational numbers. Being eventually irrational means that except for the first one OR two OR three OR ... elements the rest differs from the first AND from the second AND from the third AND ... rational number.