Suppose ${A_n} \subset X, n = 1,2,...$, show that $\cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i}) = A_n$?

Question

Suppose ${A_n} \subset X, n = 1,2,\ldots$, let $$B = \{\cap_{i=1}^{\infty} A_i^{e_i}: e_i = 1 \text{ or } -1\},$$ where $A_n^1 = A_n$, $A_n^{-1} = A_n^\complement = X \setminus A_n$. How to show that $$\cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i}) = A_n?$$

If $n = N$ is finite, for example $N=3$, then $\cup_{e_1 = 1}(\cap_{i=1}^{3} A_i^{e_i}) = A_1$ can be proved by $$(A_1A_2A_3^\complement \cup A_1A_2A_3) \cup (A_1A_2^\complement A_3 \cup A_1A_2^\complement A_3^\complement) = A_1A_2 \cup A_1A_2^\complement = A_1.$$ I don't know whether this way can be applied to prove in $N = \infty$ case since $\cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i})$ may have uncountable union, i.e. $\{\cap_{i=1}^{\infty} A_i^{e_i}: e_n = 1\}$ may have uncountable members.

Any hint?

@Randall I had a typo. I edit the question.Here $A_n \subset X$. Actually this question is some part of proof that infinite sigma-field has cardinality at least that of continuum — Spaceship222, Jul 31 '19 at 15:09
@Randall We need to specify $X$ in order to know what $A^c$ is... — David C. Ullrich, Jul 31 '19 at 15:09
Is the indexing set (e.g. $I$ for ${A_i}_{i\in I}$) countable? — R. Burton, Jul 31 '19 at 15:28
@R.Burton my question(and the notation) is from second answer of this question https://math.stackexchange.com/questions/320035/if-s-is-an-infinite-sigma-algebra-on-x-then-s-is-not-countable — Spaceship222, Jul 31 '19 at 15:39
@R.Burton I think product as in product of sets (intersection). — AlvinL, Aug 06 '19 at 07:36

score 1 · Accepted Answer · answered Aug 16 '19 at 01:41

Here is my solution:
Clearly $\cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i}) \subset A_n$.$\forall x \in A_n, x \in A_i \text{ or } A_i^\complement \ ,\forall i \neq n$,then there is one(and only one) realization of $\{e_i\}$ with $e_n = 1$ such that $x \in \cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i})$, thus $A_n \subset \cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i})$

Suppose ${A_n} \subset X, n = 1,2,...$, show that $\cup_{e_n = 1}(\cap_{i=1}^{\infty} A_i^{e_i}) = A_n$?

1 Answers1