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I have studied the proof of Monotone Class Theorem from John B. Walsh's Knowing the Odds. I feel that a part of the proof is an overkill. I'm attaching the proof below.

The author has defined $\mathcal{G'}$ to be the smallest monotone class containing $\mathcal{F_0},$ but then defined the class $\mathcal{C_1}$ to show that $\mathcal{F_0} \subset \mathcal{C_1}=\mathcal{G'}$ and hence $\mathcal{G'}$ contains $\mathcal{F_0}$. If $\mathcal{G'}$ is defined to contain $\mathcal{F_0},$ why do we need to prove that again$?$

Any clarification on whether it is really an overkill or I'm just missing something would be much appreciated. Thank you.

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The role of the sets $\mathcal C_1$ and $\mathcal C_2$ is not to show that $\mathcal G'$ contains $\mathcal F_0$; that is immediate from the definition of $\mathcal G'$, as you have noted. Their role is to show that $\mathcal G'$ is closed under finite intersections, which is the relevant remark in the second sentence you highlight.

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  • $\begingroup$ Thanks. So we need $\mathcal{C_1}=\mathcal{G'}$ to show that $\mathcal{C_2}$ actually contains $\mathcal{F_0}$. Otherwise we can't say that $\mathcal{C_2} \supset \mathcal{G'}$. $\endgroup$
    – user405743
    Commented Jul 16, 2017 at 16:22

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