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The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets by making the $\sigma$-algebra small, so here are two ways of making them big (either seems interesting to me):

  1. For every set $X$ there exists a non-trivial measure on the entire power set of $X$. (Here non-trivial means that points have measure zero.)

  2. For every set $X$, the completion of every probability measure on a set $X$ contains the entire power set of $X$.

Are there counterexamples to either of these already in ZF? If not, what is their consistency strength? Are either compatible with other axioms set theorists find attractive, such as dependent choice or the axiom of determinacy?

Edit James Hanson points out I need to exclude countable sets.

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    $\begingroup$ What about subsets of a countable set? You're not going to get a non-trivial countably additive measure there. Similarly if you require a fairly natural homogeneity condition (every subset of non-maximal cardinality has measure zero) then you'd have problems with countable cofinality too. $\endgroup$
    – James E Hanson
    Commented May 1, 2019 at 12:39
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    $\begingroup$ The restriction of (1) to uncountable sets can hold. Assume every set of reals is Lebesgue measurable, $\omega_1$ is measurable, and every uncountable set admits an injection from either $\omega_1$ or $\mathbb R$. (These conditions hold in $L(\mathbb R)$ assuming $\text{AD}^{L(\mathbb R)}$.) Then there is a nontrivial measure on $P(X)$ for every uncountable set $X$, namely the pushforward of the measure on $P(\omega_1)$ or $P(\mathbb R)$ by an injection into $X$. $\endgroup$
    – Gabe Goldberg
    Commented May 1, 2019 at 15:59
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    $\begingroup$ For (2), I assume you're only referring to completions of non-trivial measures (otherwise this fails for $X=2$). In which case, this can hold only vacuously, namely in a model where the closure of the singletons under countable union is the whole universe, e.g. Gitik's model. I suspect for both (1) and (2), what you are really interested in is measures that generalize Lebesgue measure, rather than arbitrary atomless probability measures. $\endgroup$
    – Elliot Glazer
    Commented May 1, 2019 at 21:40
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    $\begingroup$ @RobertFurber I posted this as a comment because I think arsmath may want to modify the nontriviality condition in (1) to rule out the sort of example I have given. If this is not the case, I am happy to post my comment as an answer. $\endgroup$
    – Gabe Goldberg
    Commented May 1, 2019 at 23:18
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    $\begingroup$ @arsmath As you see from Gabe's comment, you probably want to strengthen the nontriviality condition in a way that relates the size of $X$ and, say, the completeness of the measure. This requires some care when $X$ is nonwell-orderable. $\endgroup$
    – Andrés E. Caicedo
    Commented May 2, 2019 at 1:02

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