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I am new to measure theory and real analysis and am trying to double check my understanding of monotone classes.

My question:

Can monotone classes be finite? (It is not clear to me whether the idea of increasing or decreasing sets refers to STRICTLY increasing or decreasing sets.)

A related question:

Is any subset of a monotone class itself a monotone class? (The reason I ask is that I do now know the answer to the previous question.)

Thanks in advance.

1 Answers 1

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The definition of monotone class refers to not-necessarily-strictly increasing or decreasing sequences of sets.

An example of a finite monotone class of a set $X$ is $\{\varnothing\}$.

An example of a subset of a monotone class that is not itself a monotone class is the subset $\{\text{finite subsets of }\mathbb{N}\}\subset\mathcal{P}(\mathbb{N}).$ The power set of $\mathbb{N}$, $\mathcal{P}(\mathbb{N})$, is certainly a monotone class, but the increasing sequence of sets $\varnothing\subset\{1\}\subset\{1,2\}\subset\cdots$ each of which lies in the set $\{\text{finite subsets of }\mathbb{N}\}$, has a union of $\mathbb{N}$, which is not in that set.

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    every finite class is monotone class....2016-11-10