[FOM] Measures on Arbitrary Sets of Reals

[Dmytro Taranovsky]

As a consequence of the axiom of choice, there is no 
translation-invariant countably additive measure on R that extends the 
standard measure (or is otherwise non-atomic).  This leads to the 
question:  Which properties can a total extension of the measure have?

There is also (a somewhat questionable) metaphysical argument for 
existence of the preferred total measure on R^n (at least for bounded 
sets), and we can ask which properties such measure would satisfy.  The 
argument is that the measure of a subset S of [0,1] is the probability 
that a random number from [0,1] belongs to S (and similarly with 
[0,1]^n).  It uses two premises: (1) It is (physically) possible to pick 
a random real number from the unit interval, and (2) In a random 
process, every fixed predicate has a probability (even if the predicate 
is a pathological set of real numbers that cannot be defined).  (A 
metaphysical possibility of random numbers is needed to make probability 
a prior notion rather than something derived from probability measure 
spaces.)

It is consistent with ZFC (relative to a measurable cardinal) that there 
is total countably-additive measure on R that extends the standard 
measure, but it cannot be translation-invariant, and its existence is 
inconsistent with the Continuum Hypothesis.  I think that 
countable-additivity is too much to ask for measures of arbitrary sets 
of reals.  Key concepts, including that of Lebesgue integral, make sense 
based on just finite additivity, although some properties depend on 
countable additivity.  Also, for any notion of picking a random natural 
number, the corresponding probability measure would only be finitely 
additive.

Provably in ZFC there is a translation-invariant finitely additive total 
measure on R^n that extends the standard measure.

I am not sure whether we can also require scale invariance (mu(X*k) = 
k^n*mu(X)).  I am also unsure whether we can require compatibility with 
products:  mu(X x Y) = mu(X)*mu(Y) where mu(X) and mu(Y) are finite (in 
each R^m, mu is m dimensional measure).

The situation with rotations is more complicated:  As Banach-Tarski 
paradox demonstrates, we cannot allow general rotation invariance (for 
n>2).  There are 3 choices as to which invariance to require.
A. Invariance under a finite group of rotations and reflections.  If we 
are motivated by direct probability considerations, then by a symmetry 
argument (specifically, it should not matter in which order we pick n 
random real numbers; and another argument for axis reversals) this 
appears to be the preferred option, with symmetry under permutations and 
reversals of the axes (like a hypercube).  (Note: Under probability 
considerations, we would also want compatibility with products, and, at 
least for bounded sets, for each of the axes, scale invariance.)
B. Invariance under an infinite amenable (in the discrete topology) 
group of rotations and reflections.  For R^3, this allows a measure to 
also be invariant for rotations around a specific (for example z) axis, 
plus reflections through the origin and reflections through the axis.  
However, making rotations around z-axis special contradicts our 
intuitions about isotropy of space.
C.  Weaken the notion of rotation invariance.  For example, for a 
measure mu, for every rotation U and every formula phi with real 
parameters and one free variable, require phi(mu) <--> phi(U[mu]) where 
U[mu](X) = mu(UX).  I am not sure if this is consistent.  I am also not 
sure whether at the price of loss of translation invariance, rotations 
can be extended to the set of all definable from real parameters 
Lebesgue measure preserving bijective functions f: R^n->R^n.  Assuming 
that is consistent, that may be the most natural theory for arbitrary 
measures.

Finally, while a non-atomic total measure on R is not expected to be 
ordinal definable, there might be a (set theoretically) definable 
complete theory of (for example) third-order arithmetic augmented with a 
total finitely-additive measure mu that is (in a certain sense) 
preferable to all other theories of third order arithmetic with mu.

Dmytro Taranovsky