I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\mu_n$ on $\mathbb R^n$ that say, intuitively, for each subset of $[0, 1)^n$ for which the question makes sense, what is the chance that a random point in $[0, 1)^n$ falls inside this set.
This was partly answered by this answer, which says that if there exists a total extension $m$ of the Lebesgue measure (which is true if real-valued measurable cardinals exist) then $m\otimes m$ satisfies what I call the Null-Cavalieri principle w.r.t. the Lebesgue measure on $\mathbb R$ on both axes, which states that for each set $X\subset\mathbb R^2$, if for all $x\in\mathbb R$, $\{y\in\mathbb R|(x, y)\in X\}$ is a null set, then $(m\otimes m)(X)=0$, and the same is true if $\{x\in\mathbb R|(x, y)\in X\}$ is a null set for each $y$. If we accept the existence of such a measure as an axiom, Freiling's axiom follows, and similarly does the stronger claim that $\text{non}(\mathcal N)<\frak c$, i.e. there is a non-Lebesgue-measurable set of cardinality less than the continuum, which also contradicts Martin's axiom.
However, if I accept the intuitiveness of this claim, I can't see a reason not to require more about the measures that fill the intuitive notion of "chance that a random point in the cube is in a set", such as the following:
- Invariance under operations: $m\otimes m$ is already invariant under the reflection $(x, y)\mapsto(y, x)$, and if $m$ is replaced by the measure $m'$ that satisfies $m'(X)=\frac12(m(X)+m(\{-x|x\in X\}))$ then it's also invariant under reflections along the axes, and so under $90^\circ$ rotations around the origin. Also, it can be restricted only to the sets all of whose translations have the same measure without hurting the Null-Cavalieri principle, thereby making the measure invariant to translations, and also to $90^\circ$ rotations around any point. However, I intuitively think that the probability a point in a $1\times 1$ square lands in a set shouldn't change if we look at the plane differently and draw a different square around the same set, and so I intuitively want to require rotation invariance as well, and similarly I think that applying any linear transformation $T$ to a set should multiply its measure by $|\det T|$.
- Extension to higher dimensions: The same way there is an extension of the Lebesgue measure to subsets of the plane that satisfies the Null-Cavalieri principle w.r.t. the Lebesgue measure on the line, I want to require the existence of an extension (not necessarily a total one) $\mu_n$ of the Lebesgue measure on each $\mathbb R^n$ so that the Null-Cavalieri principle is satisfied among them. Note that if they are all invariant to axis permutations (or to $90^\circ$ rotations around the axes) one can conclude the Null-Cavalieri principle w.r.t. all axes from the Null-Cavalieri principle w.r.t. one axis, i.e. that for each set $X\subset\mathbb R^{n+1}$ if for all $x\in\mathbb R$, $\mu_n(\{(x_1, x_2,\dots, x_n)\in\mathbb R^n|(x_1, x_2,\dots, x_{n-1},x)\in X\})=0$ then $\mu_{n+1}(X)=0$. Also note that this implies the existence of a limit cardinal between $\text{non}(\mathcal N)$ and $\frak c$.
- The same intuition for the Null-Cavalieri principle makes me also conclude this version of Cavalieri's principle: for each pair of sets $X, Y\subset \mathbb R^{n+1}$ if for all $x\in\mathbb R$, $\mu_{n}(\{(x_1, x_2,\dots, x_{n})\in\mathbb R^{n}|(x_1, x_2,\dots, x_{n},x)\in X\})=\mu_{n}(\{(x_1, x_2,\dots, x_{n})\in\mathbb R^{n}|(x_1, x_2,\dots, x_{n},x)\in Y\})$ and $X$ is measurable according to $\mu_{n+1}$ then $\mu_{n+1}(Y)=\mu_{n+1}(X)$. Like before, the condition listed here is sufficient in the case that the measures are invariant to axis permutations, but if not then I want to explicitly require other versions, one along each axis. Note that this version of Cavalieri's principle doesn't follow from the conclusion of Fubini's theorem Wikipedia mentions, since that one only deals with the areas under measurable functions while in this version it is not assumed that $Y$ is the area under a function and the fact that it's measurable is also not assumed but concluded.
- It's also intuitive for me that if a set in $[0, 1)^2$ has all the rows either full or empty, it shouldn't matter if I randomly choose a point in $[0, 1)^2$ and check if it's in the set or choose a row from $[0, 1)$ and check if it is full, or put it another way, if I shoot 2 arrows on the segment $[0, 1)$ then the fact that I shot the second arrow shouldn't change the probability of claims about the location of the first arrow. More formally and generally, I want to require that for a set $X\subset\mathbb R^n$, if $X\times[0, 1)$ is measurable according to $\mu_{n+1}$ then $X$ is measurable according to $\mu_n$ with the same measure. Again, if the measures aren't invariant under axis permutations more versions are required. Note that the linked example doesn't satisfy this, since if, for example, $\frak c$ is the least real-valued measurable, then every $X\subset\mathbb R$ of cardinality less than $\frak c$ has $(m\otimes m)(X\times I)=0$, but trying to extend $\mu_1$ to include all such sets gives a contradiction to the Null-Cavalieri principle in the same way as Freiling's argument. Assuming the existence of smaller real-valued measurable cardinals sounds like it might fix this problem, but I don't know how. Note that if one replaces the assumption that $X\times[0, 1)$ is measurable by the assumption that $X\times X$ is measurable (and concludes that $X$'s measure is the square root of its measure), the smallest set not measurable according to $\mu_1$ (i.e. one with the least cardinality) already gives you a contradiction.
Is the existence of measures that satisfy these conditions (or some large subset of them) consistent with ZFC (allowing common assumptions like large cardinals)? Is there a small subset of these conditions that already causes a contradiction?
Thanks!
