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From: OsherD on 28 Dec 2009 01:54
From Osher Doctorow

The last post indicates a question as to whether bounded sets can
exceed or equal some unbounded sets in probability even if probability
is proportional to normalized volumes (Lebesgue measures) of sets when
defined.

The answer appears to be "yes", but it may not quite be an anomaly or
paradox. We have already seen that in many respects, bounded and
unbounded sets go together. What I've called A and B could actually
be labelled A and B ' , in which case B ' could have P(B ' ) > 1/2 .

Even without this relabelling, to refer to A and/or B as bounded
should probably be interpreted as P(A) and/or P(B) < 1/2 or < = 1/2 in
a normalized Probability sense in an unbounded Universe. If it does
not follow such a condition, then we simply have to keep track of
which sets are bounded or unbounded from physical considerations.
This is often not difficult.

An example of a bounded Universe is the "Universe of a coin landing on
heads (one side) or tails (the other side)", which with homogeneous
coins thrown without "bias" is 1/2 for each side. The whole Universe
has probability 1, and there are only two events, heads or tails, each
with probability 1/2, and each bounded.

Our physical Universe may be an unbounded Universe, in which case we
assign a probability of 1 to the whole Universe, and 1/2 can be
considered the "default" upper bound or least upper bound of bounded
set/events, but one which can be violated for Effects when Memory is
involved, though presumably not for Causes.

The real line R is an example of a mathematically unbounded Universe.
So is the "real" plane, R^2, R^3, R^4, and so on.

Osher Doctorow
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