Characteristic Function of Limit Superior of Sequence of Sets

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Theorem

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

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


Then:

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

where:

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


Proof


Sources