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 $x \in [0,1)$ then $(1_{A}+1_{B}\leq x ) = (A \cup B)^{\complement} \in \mathbb{F} $
if $x \in [1,2) $ then $ (1_{A}+1_{B} \leq x ) = (A \cap B )^{ \complement } \in \mathbb{F} $
if $x \geq 2 $ 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:
- Do you know a counterexample to this proposition?
- Do you have an idea that could help me to prove it?
Thanks in advance.