lebesgue

Another fantastic book on Lebesgue Integration/Measure theory is Lebesgue Integration in Euclidean Spaces by Frank Jones.

In analysis, people generally mean Lebesgue integrable unless otherwise specified.

The Lebesgue measure is implied on the Reals if none is given, and the computable reals have measure zero.

Some sorts of random walks inside this bounding box will converge to this scaled Lebesgue measure.

The Lebesgue measure could be a candidate for the measure you are asking for.

To imagine he invented Lebesgue measure theory on his own in his late teens.

My buddies Riemann & Lebesgue had quite a different story on the matter...

Analysis - The Elias Stein, Rami Shakarchi Analysis series, Meassure Theory by Terrence Tao, Lebesgue Integration on Euclidean Spaces by Frank Jones, A radical approach to Lebesgue Integration by Bressoud 2.

I sometimes felt dumb studying Lebesgue integration and transfinite induction, but I never felt lost like I did in DiffyQ.

But, also he goes ahead and shows that a function has a Riemann integral if and only if it is continuous everywhere except on a set of Lebesgue measure 0, without really saying much about Lebesgue measure.

We can't reasonably talk about a percent coverage, since the Lebesgue measure of the reals is infinite, but as a non-technical description, 'zero percent' is morally equivalent to saying it only covers a measure-zero set.

I.e., it should be uniform with respect to the usual Lebesgue measure on the sphere.

For a quick intro to Lebesgue integration you can read the beginning of Rudin's "Real and Complex Analysis" or Halsey Royden's "Real Analysis".

And in particular: You talk about "creating a bounding box" for the reference Lebesgue measure -- saying that this first step is easy is basically assuming the whole problem is solved.

Let's clean up the "controversy": There's some pure math due to Lebesgue, Kolmogorov, and others that defines, clearly, precisely, a 'probability space' along with trials, events, probabilities, and random variables.

"with probability 1" is a fairly standard phrasing that is equivalent to infinitesimally-small probability (e.g., a more precise statement might be: the set of oracles where the result is not true has "zero Lebesgue measure" -- I haven't checked to see exactly what formal statement they use).

J’avais passé outre, comme chose allant de soi - tout comme Lebesgue, quelques décennies plus tôt, avait dû passer outre.

To me the key to probability, and why it is really not the same as just Lebesgue's measure theory, is the role of independence.

So, Kolmogorov finds something in Lebesgue's theory, that is, a particular measure space that has everything we want for probability theory.