Let $(X,\mathcal{A},\mu)$ a measurable space where $X$ is a set, $\mathcal{A}$ is a $\sigma$-álgebra and $\mu:\mathcal{A}\rightarrow[0,\infty]$ is a measure. Let $(\chi\_{E\_{n}})$ be a sequence of characteristic functions; this is
$$\chi\_{E\_{n}}(x)=\begin{cases}
0,&x\notin E\_n,\\
1, &x\in E\_n. 
\end{cases}$$
where $E\_n\in \mathcal{A}$. Suppose that the sequence $(\chi\_{E\_{n}})$ is a Cauchy sequence in $L^p(X,\mathcal{A},\mu)$; that is given $\epsilon>0$ exists $n\_0\in \mathbb{N}$ such that
$$\|\chi\_{E\_n}-\chi\_{E\_{n}} \|\_p=\left(\int|\chi\_{E\_{n}}-\chi\_{E\_{m}}|^pd\mu\right)^{1/p}<\epsilon,\,\forall n\geq n\_0.$$
I would like to probe that exists a function $f\in L^p$ such that $f=\chi\_{E}$ where $E\in \mathcal{A}$ such that the sequence $(\chi\_{E\_{n}})$ converges to $f$ in $L\_p$; that is, given $\varepsilon>0$ exists $n\_1>0$ such that
$$\|\chi\_{E\_{n}}-f\|\_p<\varepsilon,\,\forall n\geq n\_1.$$
**My attempt:** I think the function is $f=\chi\_{E}$ where
$$E=\bigcup\_{m=1}^{\infty}\left(\bigcap\_{k=m}^{\infty}E\_k\right).$$
The reason I believe it is that the identity
$$|\chi\_{E\_{n}}-\chi\_{E\_{m}}|=\chi\_{E\_{n}\triangle E\_{m}}$$
where $E\_{n}\triangle E\_{m}=E\_n\backslash E\_m\cup E\_m\backslash E\_n$ together with the Cauchy property implies that
$$\mu(E\_{n}\triangle E\_{m})<\epsilon$$
So my reasoning tells me that the "substance" of the functions is in their intersection.