measure theory – Find a counterexample of: If $1_{A}+1_{B}$ is a random variable then $A$ and $B$ are measurable

I was trying to prove the following proposition, if $1_{A}+1_{B}$ is a random variable then $A$ and $B$ are measurable.

Where $1_{A}$ is the indicator, function given by

begin{equation}
1_{A}(x) =
begin{cases}
1, x in A \
0, x notin A
end{cases}
end{equation}

I proved that the converse is true. Here’s the proof:

Let $A$ and $B$ be measurable sets, i.e, $A$, $B in mathbb{F}$, where $mathbb{F}$ is a $sigma$-algebra. Let $x in mathbb{R}$,

If $x<0$ then $(1_{A}+1_{B}leq x ) = emptyset in mathbb{F} $

if $x=0$ then $(1_{A}+1_{B}leq x ) = A^{complement} cap B^{complement} in mathbb{F} $

if $x in {1,2} $ then $ (1_{A}+1_{B} leq x ) = (Acup B)cup (A^{complement}cap B^{complement} ) in mathbb{F} $

if $x geq 3 $ then $ (1_{A}+1_{B} leq x ) = Omega in mathbb{F} $

This implies that $1_{A}+1_{B}$ is random variable.

But not able to prove that if $1_{A}+1_{B}$ is a random variable then $A$ and $B$ are measurable. I don’t know where to start.

I have the following Questions:

  1. Do you know a counterexample to this proposition?
  2. Do you have an idea that could help me to prove it?

Thanks in advance.