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==== Hewitt-Savage 0-1 Law ====

Let \((X_i)_{i\in\mathbb{N}}\) be a family of **independent and identically distributed (i.i.d.)** random variables defined on a common probability space and taking values in a measurable space \((\mathsf{X}, \mathcal{X})\).

**Equivalent Formulation:**
We can also consider \(X_i\) as the coordinate projection associated with the probability space \((\mathsf{X}^\mathbb{N}, \mathcal{X}^{\otimes\mathbb{N}}, \mathbb{P})\), where, under \(\mathbb{P}\), the sequence \((X_i)\) is **i.i.d**.

**Permutation-Invariant σ-Fields:**
For any \(n \in \mathbb{N}\), let \(\mathcal{G}_n\) be the σ-field generated by measurable functions \(f: \mathsf{X}^\mathbb{N} \to \mathbb{R}\) that are invariant under any permutation of the first \(n\) coordinates. Define the σ-field $\mathcal{G}_\infty$ as:
\[
\mathcal{G}_\infty = \bigcap_{n\in\mathbb{N}} \mathcal{G}_n,
\]
which is generated by measurable functions invariant under any permutation of a finite number of coordinates.

<WRAP center round tip 80%>
**Statement:**
For any \(A \in \mathcal{G}_\infty\), \(\mathbb{P}(A) = 0\) or \(1\). 

This result is known as the **Hewitt-Savage 0-1 Law**.

</WRAP>


==== Proof of the Hewitt-Savage 0-1 Law ====

**Objective:**
Show that \(\mathbb{P}(A) = \mathbb{P}(A)^2\), which implies \(\mathbb{P}(A) \in \{0,1\}\).

**Key Idea:**
We aim to establish the identity \(\mathbb{P}(A) = \mathbb{E}[\mathsf{1}_A \mathsf{1}_A] = \mathbb{E}[\mathsf{1}_A]^2 = \mathbb{P}(A)^2\). To do so, the idea is to approximate $\mathsf{1}_A$ by the indicators of two different independent events. 

**Step 1: Approximation Lemma**
Let \(\delta > 0\). By the [[world:approximation-lemma|Approximation Lemma]], there exist \(n \in \mathbb{N}\) and a set \(B \in \mathcal{F}_n = \sigma(X_{1:n})\) such that:
\[
\mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_B|\right] \leq \delta.
\]
Since \(B \in \mathcal{F}_n\), there exists \(\bar{B} \in \mathcal{X}^{\otimes n}\) such that:
\[
\mathsf{1}_B = \mathsf{1}_{\bar{B}}(X_{1:n}).
\]
Thus, we have:
\[
\mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_{\bar{B}}(X_{1:n})|\right] \leq \delta.
\]


**Step 2: Permutation Argument**
Define a permutation \(\pi\) on \(\{1, \ldots, 2n\}\) that swaps the first \(n\) coordinates with the next \(n\) coordinates:
\[
\pi(1:2n) = (n+1:2n, 1:n).
\]
Since \(A \in \mathcal{G}_\infty\), the event \(A\) is invariant under \(\pi\). Therefore:
\[
\delta \geq \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_B|\right] = \mathbb{E}\left[|\mathsf{1}_A(X_{\pi(1:2n), 2n:\infty}) - \mathsf{1}_B(X_{\pi(1:2n), 2n:\infty})|\right] = \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_{\bar{B}}(X_{n+1:2n})|\right].
\]



**Step 3: Independence and Expectation**
Introduce the intermediate quantities \(\mathsf{1}_{\bar{B}}(X_{1:n})\) and \(\mathsf{1}_{\bar{B}}(X_{n+1:2n})\) which are independent by independence of \((X_i)\). Then, we obtain:
$$
|\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_A]\mathbb{E}[\mathsf{1}_A]| \leq |\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_{\bar{B}}(X_{1:n})\mathsf{1}_{\bar{B}}(X_{n+1:2n})]| + | \mathbb{E}[\mathsf{1}_{\bar{B}}(X_{1:n})]\mathbb{E}[\mathsf{1}_{\bar{B}}(X_{n+1:2n})]- \mathbb{E}[\mathsf{1}_A] \mathbb{E} [\mathsf{1}_A]| \leq 4\delta.
$$
Since \(\delta > 0\) is arbitrary, we conclude that:
\[
\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] = \mathbb{E}[\mathsf{1}_A]^2,
\]
which implies \(\mathbb{P}(A) = \mathbb{P}(A)^2\). Therefore, \(\mathbb{P}(A) \in \{0,1\}\).


**Conclusion:**
The Hewitt-Savage 0-1 Law states that any permutation-invariant event in the tail σ-field \(\mathcal{G}_\infty\) has probability 0 or 1.
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