Characteristic Function of Limit Superior of Sequence of Sets

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 $\chi_{E_n}$

Sources

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