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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 tail σ-field 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.

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 \(\mathbb{P}(A) = \mathbb{E}[\mathsf{1}_A \mathsf{1}_A] = \mathbb{E}[\mathsf{1}_A]^2 = \mathbb{P}(A)^2\).

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Step 1: Approximation Lemma Let \(\delta > 0\). By the 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: \[ \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_{\bar{B}}(X_{1:n})|\right] \leq \delta. \]

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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: \[ \delta \geq \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_B|\right] = \mathbb{E}\left[|\mathsf{1}_A(X_{\pi(1:2n), 2n:\infty}) - \mathsf{1}_B(X_{\pi(1:2n), 2n:\infty})|\right] = \mathbb{E}\left[|\mathsf{1}_A - \mathsf{1}_{\bar{B}}(X_{n+1:2n})|\right]. \]

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Step 3: Independence and Expectation Introduce the intermediate quantities \(\mathsf{1}_{\bar{B}}(X_{1:n})\) and \(\mathsf{1}_{\bar{B}}(X_{n+1:2n})\). By the independence of \((X_i)\) and the invariance of \(A\), we obtain: \[ |\mathbb{E}[\mathsf{1}_A \mathsf{1}_A] - \mathbb{E}[\mathsf{1}_A]\mathbb{E}[\mathsf{1}_A]| \leq 4\delta. \] 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.