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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.

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1 Answer 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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Very nice clarification, thanks. –  Gandhi Viswanathan Jun 14 '12 at 19:46
    
Thanks, glad to help! –  Zev Chonoles Jun 14 '12 at 19:48
    
It seems this is all second nature and fairly obvious to mathematicians like you. Do you have any advice on how to develop this insight? Is it simply a matter of spending a few thousand hours reading and doing exercises? Or is there more to it? –  Gandhi Viswanathan Jun 14 '12 at 19:53
    
I think that the core of how to do these kinds of problems is not difficult itself, but that even when explained, it usually doesn't really sink in until one's done enough problems utilizing it. In fact, I would say that about most things in mathematics - the essential strategy is fairly comprehensible to a mathematical journeyman (or even novice), but the ability to implement the strategy quickly, accurately, and completely is something that is best gained through practice. –  Zev Chonoles Jun 14 '12 at 20:46
    
Here, I guess I would say the essential strategy I was using was just one of constructing something minimal; taking something you know works, and breaking exactly the thing necessary to make it not work. Of course, familiarity with a wide array of examples makes it easier to do this. You might be interested in reading Polya's How to Solve It, or Krantz's A Mathematician Comes of Age, about problem-solving strategies, and the notion of mathematical maturity, respectively. –  Zev Chonoles Jun 14 '12 at 20:47

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