Definition:Exhausting Sequence of Sets
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Definition
Let $S$ be a set.
Let $\SS = \powerset S$ be the power set of $S$.
Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that:
- $(1): \quad \forall k \in \N: S_k \subseteq S_{k + 1}$
- $(2): \quad \ds \bigcup_{k \mathop \in \N} S_k = S$
Then $\sequence {S_k}_{k \mathop \in \N}$ is an exhausting sequence of sets (in $\SS$).
That is, it is an increasing sequence of subsets of $S$, whose union is $S$.
It is common to write $\sequence {S_k}_{k \mathop \in \N} \uparrow S$ to indicate an exhausting sequence of sets.
Here, the $\uparrow$ denotes a limit of an increasing sequence.
Also see
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.2$