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Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.

Q. For which cardinal numbers $\alpha$ the following holds?

$$Card(X)=\alpha \Longrightarrow P(X\times X)=P(X)\otimes_{\sigma}P(X)$$

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    $\begingroup$ See Question 2 here: mathoverflow.net/q/39882/454 and the answer by Michael Greinecker $\endgroup$
    – Gerald Edgar
    Commented May 16, 2018 at 17:39
  • $\begingroup$ @Gerald Edgar: Thanks. Is it unknown when $X$ has cardinality $\mathfrak{c}$ yet? $\endgroup$
    – ABB
    Commented May 16, 2018 at 18:23
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    $\begingroup$ See Martin's answer to the same post. $\endgroup$
    – Gerald Edgar
    Commented May 16, 2018 at 23:59

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