Crimboi [14] 开门红(?)

题目背景

Rudin Real and Complex Analysis第一题, 真是开门红.

题面

We call a pairing (X,\mathfrak{M}) \sigma-algebra if \mathfrak{M} is a subset of P(X) satisfying the three properties below:

(1) X\in\mathfrak{M}

(2) If A\in\mathfrak{M}, A^c\in\mathfrak M

(3) If \lbrace A_i\rbrace_{i\in\mathbb N}\subset\mathfrak M, then \bigcup\limits_{i\in\mathbb N}A_i\in\mathfrak M

Question: Is there a \sigma-algebra (X,\mathfrak M) such that |\mathfrak M|=|\mathbb N|?

先想再看提示哦

提示

Everyone knows we should construct a set of cardinality |P(\mathbb N)| from some subset of \mathfrak M. That's not a easy task, as \cup A_i=\cup B_i might occur often in a poorly chosen subset. However, why not consider a subset whose members are disjoint?

愿你无需看标答

标答

法克好像给了hint后太简单了