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--- world:approximation-lemma [2026/02/05 07:59]
rdouc
+++ world:approximation-lemma [2026/02/05 08:02] (current)
rdouc ↷ Page moved from mynotes:approximation-lemma to world:approximation-lemma
@@ Line 3: / Line 3: @@
 ====== 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 $\omega \in \rset^\nset$.
+Let $(\Xset^{\mathbb{N}},\Xsigma^{\otimes\mathbb{N}},\mathbb{P})$ be a probability space.
+For each $k\in\mathbb{N}$, define
+$$
+\mathcal{F}_k=\sigma(X_{1:k}),
+$$
+where $X_i$ denotes the coordinate projection on the $i$-th component, that is,
+$X_i(\omega)=\omega_i$ for $\omega\in\Xset^{\mathbb{N}}$.
 <WRAP center round tip 80%>
-**Lemma**. Any set $A\in\mathcal{X}^{\otimes\mathbb{N}}$ satisfies ** the approximation property**, that is:
+**Lemma (Approximation Lemma).**
+Any set $A\in\mathcal{X}^{\otimes\mathbb{N}}$ satisfies the **approximation property**:
 For every $\delta>0$, there exist an integer $k\in\mathbb{N}$ and a set $B\in\mathcal{F}_k$ such that
@@ Line 11: / Line 19: @@
 \mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]\leq\delta.
 $$
 </WRAP>
 ===== Proof =====
+The proof is a classical application of the **monotone class theorem**.
-The proof is a standard application of the **monotone class theorem**.
+Let $\mathcal{M}$ denote the collection of all sets $A\in\mathcal{X}^{\otimes\mathbb{N}}$ that satisfy the approximation property stated in the lemma.
+We show that $\mathcal{M}$ is a monotone class.
-Define $\mathcal{M}$ as the collection of all sets $A\in\mathcal{X}^{\otimes\mathbb{N}}$ for which the above **approximation property** (defined in the Lemma) holds.
-We verify that $\mathcal{M}$ is a monotone class.
   * **Stability under set differences.**
-Let $A_0,A_1\in\mathcal{M}$ with $A_0\subset A_1$. Then we will show $A_1\setminus A_0\in\mathcal{M}$.
+Let $A_0,A_1\in\mathcal{M}$ with $A_0\subset A_1$. We claim that $A_1\setminus A_0\in\mathcal{M}$.
-Indeed, for any sets $A_0,A_1,B_0,B_1$, the following identity holds:
+For arbitrary sets $A_0,A_1,B_0,B_1$, the following identity holds:
 $$
 \mathbb{1}_{A_1\setminus A_0}-\mathbb{1}_{B_1\setminus B_0}
@@ Line 34: / Line 39: @@
 $$
-Taking expectations and absolute values yields
+Taking absolute values and expectations yields
 $$
 \mathbb{E}\big[|\mathbb{1}_{A_1\setminus A_0}-\mathbb{1}_{B_1\setminus B_0}|\big]
@@ Line 42: / Line 47: @@
 \mathbb{E}\big[|\mathbb{1}_{A_1}-\mathbb{1}_{B_1}|\big].
 $$
-When $A_0,A_1\in\mathcal{M}$, both terms on the right-hand side can be made arbitrarily small. So, the approximation property also holds for $A_1\setminus A_0$.
+Since $A_0,A_1\in\mathcal{M}$, both terms on the right-hand side can be made arbitrarily small. Hence, the approximation property holds for $A_1\setminus A_0$.
   * **Stability under increasing limits.**
@@ Line 50: / Line 56: @@
 $$
-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 $n\in \nset$.
+For any set $B$, the triangle inequality gives
-Since $\mathcal{M}$ is a monotone class containing all $\mathcal{F}_k$, it contains
 $$
-\sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}.
+\mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]
+\leq
+\mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_{A_n}|\big]
++
+\mathbb{E}\big[|\mathbb{1}_{A_n}-\mathbb{1}_B|\big].
 $$
-This completes the proof.
-{{page>:defs}}
-====== Approximation Lemma ======
-Let $(\Xset^{\mathbb{N}},\Xsigma^{\otimes\mathbb{N}},\mathbb{P})$ be a probability space.
-For each $k\in\mathbb{N}$, define
-$$
-\mathcal{F}_k=\sigma(X_{1:k}),
-$$
-where $X_i$ denotes the coordinate projection on the $i$-th component, that is,
-$X_i(\omega)=\omega_i$ for $\omega\in\Xset^{\mathbb{N}}$.
-<WRAP center round tip 80%>
-**Lemma (Approximation Lemma).**
-Any set $A\in\mathcal{X}^{\otimes\mathbb{N}}$ satisfies the **approximation property**:
-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 classical application of the **monotone class theorem**.
-Let $\mathcal{M}$ denote the collection of all sets $A\in\mathcal{X}^{\otimes\mathbb{N}}$ that satisfy the approximation property stated in the lemma.
-We show that $\mathcal{M}$ is a monotone class.
-  * **Stability under set differences.**
-    Let $A_0,A_1\in\mathcal{M}$ with $A_0\subset A_1$. We claim that $A_1\setminus A_0\in\mathcal{M}$.
-    For arbitrary 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 absolute values and expectations yields
-    $$
-    \mathbb{E}\big[|\mathbb{1}_{A_1\setminus A_0}-\mathbb{1}_{B_1\setminus B_0}|\big]
-    \leq
-    \mathbb{E}\big[|\mathbb{1}_{A_0}-\mathbb{1}_{B_0}|\big]
-    +
-    \mathbb{E}\big[|\mathbb{1}_{A_1}-\mathbb{1}_{B_1}|\big].
-    $$
-    Since $A_0,A_1\in\mathcal{M}$, both terms on the right-hand side can be made arbitrarily small. Hence, the approximation property 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
-    $$
-    A=\bigcup_{n\geq0}A_n.
-    $$
-    For any set $B$, the triangle inequality gives
-    $$
-    \mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]
-    \leq
-    \mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_{A_n}|\big]
-    +
-    \mathbb{E}\big[|\mathbb{1}_{A_n}-\mathbb{1}_B|\big].
-    $$
-    Since $\mathbb{1}_{A_n}\uparrow\mathbb{1}_A$ pointwise, the first term converges to zero, while the second term can be made arbitrarily small because $A_n\in\mathcal{M}$. Therefore, $A\in\mathcal{M}$.
+Since $\mathbb{1}_{A_n}\uparrow\mathbb{1}_A$ pointwise, the first term converges to zero, while the second term can be made arbitrarily small because $A_n\in\mathcal{M}$. Therefore, $A\in\mathcal{M}$.
 Finally, $\mathcal{M}$ is a monotone class containing all $\mathcal{F}_k$. By the monotone class theorem, it contains

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