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The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. Math. 39, 1952.

Observe that if $m: \mathcal{P}(X) \to [0, 1]$ is a diffused (vanishes on singletons) probability measure, then $F_m: 2^{X} \to [0, 1]$ defined by $F_m(1_A) = m(A)$ is a sequentially continuous (whenever $A_n \to A$, $m(A_n) \to m(A)$) but discontinuous map (the preimage of $\{0\}$ is not closed in $2^X$).

Question: Let $\kappa$ be the least cardinal such that there is a sequentially continuous but discontinuous map from $2^{\kappa}$ to $[0, 1]$. Must $\kappa$ admit a total diffused probability measure? In modern terms, is $\kappa$ real valued measurable?

The only partial result that I know here appears in D. Choodnovsky: Sequentially continuous mappings of product spaces, Seminaire D'Analyse Fonctionnelle Ecole Polytechnique, 1977-78, Exp. no. 4, pp 1-15 where, among other things, it is shown that such a cardinal must admit an $\aleph_1$-saturated sigma ideal.

The above paper is pretty old and I couldn't find any recent survey on this problem so I am wondering about the current status of the problem. Thanks!

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    $\begingroup$ A related paper might be "Sequential continuity and submeasurable cardinals" by Balcar and Hušek. The following is taken from the review of their paper from Mathscinet: The relationship between continuity and sequential continuity is, outside the class of sequential spaces, decidely one-sided. An exception can be made for the Cantor cubes $2^κ$; call a cardinal $κ$ sequential if $2^κ$ admits a real-valued function that is sequentially continuous but not continuous; call $κ$ uniformly sequential if $2^κ$ admits a real-valued function that is sequentially uniformly continuous but not continuous. $\endgroup$
    – Mohammad Golshani
    Commented Sep 18, 2015 at 6:27
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    $\begingroup$ $\to$ finally a $g$-sequential cardinal is a $κ$ such that $Z^κ$ admits a sequentially continuous but not continuous homomorphism or, equivalently, $P(κ)$ admits a diffuse sequentially continuous submeasure $\endgroup$
    – Mohammad Golshani
    Commented Sep 18, 2015 at 6:28
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    $\begingroup$ $\to$ The first cardinals of each type are all equal and weakly inaccessible but not larger than the first real-valued measurable cardinal (equality is still an open problem). $\endgroup$
    – Mohammad Golshani
    Commented Sep 18, 2015 at 6:29
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    $\begingroup$ $\to$ The authors show that the equality persists for higher versions of these cardinals, where one replaces "countable'' by "smaller than $κ$''' in the definitions. Call $κ$ submeasurable if there is a nonzero diffuse and $κ$-continuous submeasure on $P(κ)$, where $κ$-continuous means "continuous for nets of size less than $κ$''. $\endgroup$
    – Mohammad Golshani
    Commented Sep 18, 2015 at 6:30
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    $\begingroup$ There is a 1993 survey paper by Grzegorz Plebanek in FM; section 6 is the relevant bit perhaps: matwbn.icm.edu.pl/ksiazki/fm/fm143/fm14312.pdf $\endgroup$
    – Avshalom
    Commented Sep 18, 2015 at 11:37

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An answer seems to be provided in the comments to the problem.

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  • $\begingroup$ I have added this "answer" so that readers searching for problems where they can help are not mislead. If there are better ways of doing so, feel free to edit. $\endgroup$
    – Boaz Tsaban
    Commented Jan 10, 2016 at 21:18
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    $\begingroup$ It might not be a terrible idea to write what is that answer. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 10, 2016 at 21:35
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    $\begingroup$ @AsafKaragila: I do not know the rules of MO. But I find it annoying that questions answered in the comments are marked as "unanswered". I hoped this could be a simple (even if terrible) and quick solution. I hope that perhaps someone will have the time and energy to add details. I marked this "answer" community wiki, so I do not get any credit for it. Just a service to the community, which hopefully the community can improve. $\endgroup$
    – Boaz Tsaban
    Commented Jan 10, 2016 at 22:15

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