What Is True Almost Everywhere?


I was reading a thermodynamics book (C Truesdell and S Bharatha’s The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, which is a fascinating read, for the field, and includes a number of entertaining, for the field, snipes at the stuff textbook writers put in because they’re just passing on stuff without rethinking it carefully), and ran across a couple proofs which mentioned equations that were true “almost everywhere”. That’s a construction it might be surprising to know even exists in mathematics, so, let me take a couple hundred words to talk about it.

The idea isn’t really exotic. You’ve seen a kind of version of it when you see an equation containing the note that there’s an exception, such as, \frac{\left(x - 1\right)^2}{\left(x - 1\right)} = x \mbox{ for } x \neq 1 . If the exceptions are tedious to list — because there are many of them to write down, or because they’re wordy to describe (the thermodynamics book mentioned the exceptions were where a particular set of conditions on several differential equations happened simultaneously, if it ever happened) — and if they’re unlikely to come up, then, we might just write whatever it is we want to say and add an “almost everywhere”, or for shorthand, put an “ae” after the line. This “almost everywhere” will, except in freak cases, propagate through the rest of the proof, but I only see people writing that when they’re students working through the concept. In publications, the “almost everywhere” gets put in where the condition first stops being true everywhere-everywhere and becomes only almost-everywhere, and taken as read after that.

I introduced this with an equation, but it can apply to any relationship: something is greater than something else, something is less than or equal to something else, even something is not equal to something else. (After all, “x \neq -x is true almost everywhere, but there is that nagging exception.) A mathematical proof is normally about things which are true. Whether one thing is equal to another is often incidental to that.

What’s meant by “unlikely to come up” is actually rigorously defined, which is why we can get away with this. It’s otherwise a bit daft to think we can just talk about things that are true except where they aren’t and not even post warnings about where they’re not true. If we say something is true “almost everywhere” on the real number line, for example, that means that the set of exceptions has a total length of zero. So if the only exception is where x equals 1, sure enough, that’s a set with no length. Similarly if the exceptions are where x equals positive 1 or negative 1, that’s still a total length of zero. But if the set of exceptions were all values of x from 0 to 4, well, that’s a set of total length 4 and we can’t say “almost everywhere” for that.

This is all quite like saying that it can’t happen that if you flip a fair coin infinitely many times it will come up tails every single time. It won’t, even though properly speaking there’s no reason that it couldn’t. If something is true almost everywhere, then your chance of picking an exception out of all the possibilities is about like your chance of flipping that fair coin and getting tails infinitely many times over.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

7 thoughts on “What Is True Almost Everywhere?”

  1. I don’t think I’ve read the term “true almost everywhere” before? Is it a term used in the sciences more than in mathematics? I know it annoys me when people assert that ANY number raised to the zero power is 1 and completely ignore that zero to the zero power isn’t defined. Thank you for an enlightening post!

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    1. I’d encountered “almost everywhere” first in functional analysis, if I remember rightly, and at least my impression is that it has its greatest domain there. (The concept is tied to measure theory — you can be true almost everywhere if the set of exceptions has measure zero — which is probably why it doesn’t seem to get seen much before grad school.) Wikipedia suggests the same concept is in older books abbreviated pp, for “presque partout”, but that just means the same “almost everywhere”.

      I was actually a bit surprised to encounter it explicitly mentioned in a thermodynamics book, but the authors were going for a rigor they felt wasn’t present in most books of the kind.

      And, thank you for writing back.

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  2. Nice post about one nice concept in Mathematics. “Almost everywhere” is a notion often encountered within Measure Theory, meaning that something happens except for a set of zero measure. My first meeting with it was in my Probability courses, where I was told that it mean a set of possible results of an experiment whose elements would virtually not appear on practice though. Like, say, you pick the temperature of a point at random at wonder what is the probability it will be 23.5 ºC exactly. This is certainly a possible temperature, but the probability of being that exact temperature is exactly zero, since you can only get positive probabilities for ranges of temperatures.

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    1. Thank you kindly.

      I’ve been thinking to write a bit more about measure theory, since it seems to be at a golden intersection for popular mathematics: the ideas are accessible without any advanced mathematics and possibly any equations other than “the sum of 1/2 plus 1/4 plus 1/8 plus 1/16 plus (etc) is 1”, and it gets pretty quickly and easily to some results contrary to intuition, and it underlies a lot of important higher mathematics, and for all that it doesn’t seem to be over-grazed territory for pop math writers, the way the different cardinalities of infinity are.

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