Characteristic Function of Limit Superior of Sequence of Sets

Theorem

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

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

Then:

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

where:

$\chi$ denotes characteristic function

$\ds \liminf_{n \mathop \to \infty} \chi_{E_n}$ is the pointwise limit superior of the $\chi_{E_n}$

Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 9$: Problem $9 \ \text{(i)}$