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fishturtle1

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- Homework Statement
- In older books the Borel sets are often introduced as the smallest family ##\mathcal{M}## of sets which is stable under countable intersections of decreasing and countable unions of increasing sequences of sets, and which contain all open sets ##\mathcal{O}##. Use problem 3.14 to show that ##\mathcal{M} = \mathcal{B}(\mathbb{R})##.

- Relevant Equations
- ##\textbf{Definition.}## A family ##\mathcal{M} \subset \mathcal{P}(X)## which contains ##X## and is stable under countable unions of increasing sets and countable intersections of decreasing sets

$$(A_n)_{n\in\mathbb{N}} \subset \mathcal{M}, A_1 \subset \dots \subset A_n \subset A_{n+1} \uparrow A = \bigcup_{n \in \mathbb{N}} A_n \Rightarrow A \in \mathcal{M}$$

$$(B_n)_{n\in\mathbb{N}} \subset \mathcal{M}, B_1 \supset \dots \supset B_n \supset B_{n+1} \downarrow B = \bigcap_{n\in\mathbb{N}} B_n\Rightarrow B \in \mathcal{M}$$

is called a ##\textit{monotone class}##.

##\textbf{Definition.}## Let ##\mathcal{F} \subset \mathcal{P}(X)##. We define ##m(\mathcal{F})## to be the smallest monotone class containing ##\mathcal{F}##. That is, if ##\mathcal{M}## is a monotone class containing ##\mathcal{F}##, then ##m(\mathcal{F}) \subseteq \mathcal{M}##.

Problem 3.14: There are four parts:

i) Mimic the proof of theorem 3.4. to show that there is a minimal monotone class ##m(\mathcal{F})## such that ##\mathcal{F} \subset m(\mathcal{F})##.

ii) If ##\mathcal{F}## is stable w.r.t. complements, then so is ##m(\mathcal{F})##.

iii) If ##\mathcal{F}## is stable w.r.t. intersection, then so is ##m(\mathcal{F})##.

iv) Use i), ii), iii) to prove the following: Let ##\mathcal{F} \subset \mathcal{P}(X)## which is stable under the formation of intersections and complements. If ##\mathcal{M} \supset \mathcal{F}## is a monotone class, then ##M \supset \sigma(\mathcal{F})##.

Proof: Let ##A, B \in \mathcal{O}## and ##x \in A \cap B##. Then there exists ##\varepsilon_A, \varepsilon_B > 0## such that ##B_{\varepsilon_A}(x) \subset A## and ##B_{\varepsilon_B}(x) \subset B##. Let ##\varepsilon = \min\lbrace\varepsilon_A, \varepsilon_B\rbrace##. Then ##B_\varepsilon(x) \subset A \cap B##. This shows ##A \cap B \in \mathcal{O}## i.e. ##\mathcal{O}## is stable under formation of intersections.

But ##(0, 1) \in \mathcal{O}## and ##(0, 1)^c = (-\infty, 0] \cup [1, \infty) \not\in \mathcal{O}##. So ##\mathcal{O}## is not stable under complements.

If I could show that ##m(\mathcal{O})## is a ##\sigma## algebra, then ##m(\mathcal{O}) \supseteq \sigma(\mathcal{O})##. Moreover, any ##\sigma## algebra is also a monotone class, so ##m(\mathcal{O}) \subseteq \sigma(\mathcal{O})##, which would complete the proof.Have I made a mistake on the complements part? Or do I need to choose a different generating set for ##\mathcal{B}(\mathbb{R^n})##?

But ##(0, 1) \in \mathcal{O}## and ##(0, 1)^c = (-\infty, 0] \cup [1, \infty) \not\in \mathcal{O}##. So ##\mathcal{O}## is not stable under complements.

If I could show that ##m(\mathcal{O})## is a ##\sigma## algebra, then ##m(\mathcal{O}) \supseteq \sigma(\mathcal{O})##. Moreover, any ##\sigma## algebra is also a monotone class, so ##m(\mathcal{O}) \subseteq \sigma(\mathcal{O})##, which would complete the proof.Have I made a mistake on the complements part? Or do I need to choose a different generating set for ##\mathcal{B}(\mathbb{R^n})##?