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