# 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?