Characteristic Function of Limit Inferior of Sequence of Sets

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Theorem

Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets.

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


Then:

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

where:

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


Proof


Sources