The monotone class theorem states that for any algebra of sets $\cal A$ one can construct the smallest monotone class generated by this class ${\cal M}(\cal A)$. This smallest monotone class is also the smallest sigma-algebra $\Sigma(\cal A)$ generated by $\cal A$, and ${\cal M}(\cal A)=\Sigma(\cal A)$. However, I noticed that in applications of the […]

Suppose $\mathcal F$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal F$ and $f + g$, $fg$, and $cf$ are in $\mathcal F$ whenever $f, g \in \mathcal F$ and $c \in \mathbb R$. Suppose $f \in \mathcal F$ whenever $f_n \to f$ and each $f_n \in […]

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\mathcal{C}$ be a class of subsets of $\Omega$ under finite intersections and containing $\Omega$. Let $\mathcal{B}$ be the smallest class containing $\mathcal{C}$ which is closed […]

Monotone class theorem: Let $\mathcal C$ be a class of subset closed under finite intersections and containing $\Omega$ (that is, $\mathcal C$ is a $\pi$-system). Let $\mathcal B$ be the smallest class containing $\mathcal C$ which is closed under increasing limits and by difference (that is, $\mathcal B$ is the smallest $\lambda$ system containing $\mathcal […]

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