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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 $\omega\in\Xset^{\mathbb{N}}$.
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 $$ \mathbb{E}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]\leq\delta. $$
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.
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 $A_1\setminus A_0$.
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, $A\in\mathcal{M}$.
Finally, $\mathcal{M}$ is a monotone class containing all $\mathcal{F}_k$. By the monotone class theorem, it contains $$ \sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}. $$ This completes the proof.