Lemma (Approximation Lemma). Let $A\in\mathcal{X}^{\otimes\mathbb{N}}$. Then $A$ satisfies the following approximation property:

For every $\delta>0$, there exist an integer $k\in\mathbb{N}$ and a set $B\in\mathcal{F}_k$ such that $$ \mathbb{E}_{\pi}\big[|\mathbb{1}_A-\mathbb{1}_B|\big]\leq\delta. $$

Proof. The proof is a standard application of the monotone class theorem.

Define $\mathcal{M}$ as the collection of all sets $A\in\mathcal{X}^{\otimes\mathbb{N}}$ for which the above approximation property 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 $A_1\setminus A_0\in\mathcal{M}$.

Indeed, for any 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 expectations and absolute values 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 both terms on the right-hand side can be made arbitrarily small, 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 $$ A=\bigcup_{n\geq0}A_n. $$

Then $$ \mathbb{1}_{A_n}\uparrow\mathbb{1}_A \quad \text{pointwise}. $$ 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 $$ \sigma\Big(\bigcup_{k\geq0}\mathcal{F}_k\Big)=\mathcal{X}^{\otimes\mathbb{N}}. $$ This completes the proof.