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world:approximation-lemma [2026/02/05 07:51] rdouc [Proof] |
world:approximation-lemma [2026/02/05 08:02] (current) rdouc ↷ Page moved from mynotes:approximation-lemma to world:approximation-lemma |
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| ====== Approximation Lemma ====== | ====== 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%> | <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 | 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. | \mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]\leq\delta. | ||
| $$ | $$ | ||
| - | |||
| </WRAP> | </WRAP> | ||
| ===== Proof ===== | ===== 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.** | * **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} | \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] | \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]. | \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.** | * **Stability under increasing limits.** | ||
| Line 50: | Line 56: | ||
| $$ | $$ | ||
| - | Then | + | For any set $B$, the triangle inequality gives |
| $$ | $$ | ||
| - | \mathbb{1}_{A_n}\uparrow\mathbb{1}_A \quad \text{pointwise}. | + | \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]. | ||
| $$ | $$ | ||
| - | By monotone convergence, this implies that $A$ also belongs to $\mathcal{M}$. | ||
| - | Since $\mathcal{M}$ is a monotone class containing all $\mathcal{F}_k$, it contains | + | 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 | ||
| $$ | $$ | ||
| \sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}. | \sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}. | ||
| $$ | $$ | ||
| This completes the proof. | This completes the proof. | ||
| + | |||
world/approximation-lemma.1770274314.txt.gz · Last modified: 2026/02/05 07:51 by rdouc
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