[FOM] Measures on Arbitrary Sets of Reals
Dmytro Taranovsky
dmytro at mit.edu
Mon Feb 27 15:25:22 EST 2012
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
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