Characteristic Function of Limit Inferior of Sequence of Sets

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Theorem

Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets.

Let $E := \displaystyle \liminf_{n \mathop \to \infty} \, E_n$ be the limit inferior of the $E_n$.


Then:

$\displaystyle \chi_E = \liminf_{n \mathop \to \infty} \, \chi_{E_n}$

where:

$\chi$ denotes characteristic function
$\displaystyle \liminf_{n \to \infty} \, \chi_{E_n}$ is the pointwise limit inferior of the $\chi_{E_n}$


Proof


Sources