/sci/ - Science & Math

I'm insecure about an exercise from Folland's Real Analysis book (Chapter 2, Problem 36).

Let [math](\Omega, \mathcal{F}, \mu)[/math] be a measure space, and let [math]\{E_n\}_{n \geq 1}[/math] be a sequence of measurable sets with finite measure.

Further assume that [math]\chi_{E_n} \to f \text{ in } L^1[/math]. Then [math]f[/math] agrees almost everywhere with the characteristic function of a measurable set.

Here's what I think:

We can find a subsequence of [math]\{\chi_{E_n}\}_{n \geq 1}[/math] that converges almost everywhere to [math]f[/math]. Then for every point in [math]\Omega[/math], said subsequence must be either converging towards [math]0[/math] or converging towards [math]1[/math], since the pointwise a.e. limit exists.

How do I make this proof less handwavy and more formal? Is it correct? Thanks in advance.