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--- world:approximation-lemma [2026/02/05 08:01]
rdouc
+++ world:approximation-lemma [2026/02/05 08:02] (current)
rdouc ↷ Page moved from mynotes:approximation-lemma to world:approximation-lemma
@@ Line 1: / Line 1: @@
-{{page>:defs}}
-====== Approximation Lemma ======
-Let $(\Xset^\nset,\Xsigma^{\otimes \nset}, \PP)$ be a probability space. We write $\mathcal{F}_k=\sigma(X_{1:k})$ where $X_i$ is the coordinate projection on the $i$-th component: $X_i(\omega)=\omega_i$ where $\omega \in \rset^\nset$.
-<WRAP center round tip 80%>
-**Lemma**. Any set $A\in\mathcal{X}^{\otimes\mathbb{N}}$ satisfies ** the approximation property**, that is:
-For every $\delta>0$, there exist an integer $k\in\mathbb{N}$ and a set $B\in\mathcal{F}_k$ such that
-$$
-\mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]\leq\delta.
-$$
-</WRAP>
-===== Proof =====
-The proof is a standard application of the **monotone class theorem**.
-Define $\mathcal{M}$ as the collection of all sets $A\in\mathcal{X}^{\otimes\mathbb{N}}$ for which the above **approximation property** (defined in the Lemma) holds.
-We verify that $\mathcal{M}$ is a monotone class.
-  * **Stability under set differences.**
-Let $A_0,A_1\in\mathcal{M}$ with $A_0\subset A_1$. Then we will show $A_1\setminus A_0\in\mathcal{M}$.
-Indeed, for any sets $A_0,A_1,B_0,B_1$, the following identity holds:
-$$
-\mathbb{1}_{A_1\setminus A_0}-\mathbb{1}_{B_1\setminus B_0}
-=\mathbb{1}_{A_1}\mathbb{1}_{A_0^c}-\mathbb{1}_{B_1}\mathbb{1}_{B_0^c}
-=\mathbb{1}_{A_1}(\mathbb{1}_{A_0^c}-\mathbb{1}_{B_0^c})
-+(\mathbb{1}_{A_1}-\mathbb{1}_{B_1})\mathbb{1}_{B_0^c}.
-$$
-Taking expectations and absolute values yields
-$$
-\mathbb{E}\big[|\mathbb{1}_{A_1\setminus A_0}-\mathbb{1}_{B_1\setminus B_0}|\big]
-\leq
-\mathbb{E}\big[|\mathbb{1}_{A_0}-\mathbb{1}_{B_0}|\big]
-+
-\mathbb{E}\big[|\mathbb{1}_{A_1}-\mathbb{1}_{B_1}|\big].
-$$
-When $A_0,A_1\in\mathcal{M}$, both terms on the right-hand side can be made arbitrarily small. So, the approximation property also holds for $A_1\setminus A_0$.
-  * **Stability under increasing limits.**
-Let $(A_n)_{n\geq0}$ be an increasing sequence of sets in $\mathcal{M}$, and define
-$$
-A=\bigcup_{n\geq0}A_n.
-$$
-Then using the triangular inequality for any set $B$, $\PE[|\mathbb{1}_A-\mathbb{1}_B|] \leq \PE[|\mathbb{1}_A-\mathbb{1}_{A_n}|] + \PE[|\mathbb{1}_{A_n}-\mathbb{1}_B|]$, we can easily deduce that $A$ also belongs to $\mathcal{M}$ provided that $A_n\in \mathcal{M}$ for any $n\in \nset$.
-Since $\mathcal{M}$ is a monotone class containing all $\mathcal{F}_k$, it contains
-$$
-\sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}.
-$$
-This completes the proof.
 {{page>:defs}}

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