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Measure theory is not about events of probability zero, it is about how to ignore them and prevent them from messing up results. Let's say we play a game, we u
by afiori 2y ago
Measure theory is not about events of probability zero, it is about how to ignore them and prevent them from messing up results.
Let's say we play a game, we uniformly pick a random real between 0 and 1 and you win if it is rational, I win if it is not.
You can obviously see that it is unfair, but how do you prove it? You need a concept of integration that can easily ignore a dense set of discontinuities, Riemann integration is not going to give you any good results (in this case at the very best it would tell you that you win between 0% and 100% of the times, not very useful)
Measure theory and Lebesgue integration are a way to discard this noise.
Actually in measure theory you generally use L¹, L², etc spaces where functions are defined modulo null sets; that is the function that is 1 on rational and 0 on irrationals is considered to be the same function as the constant 0.
In measure theory on the Reals the value of a function at a specific point is generally considered irrelevant.
- ajkjk 2y agoThe difference is that I'm interested in a version of physics and economics that is not aware of the distinction between 'real' and 'rational'. Hence none of that should make a difference.
- afiori 2y agoIf you work with mostly continous and/or well behaved functions you do not need most of these. I suspect that in both physics and economics you might end up using stochastic methods that use concepts and techniques similar to those of measure theory
- monadINtop 2y agothen you are interested in a version of physics that doesn't exist, even though some might choose to supress the details at low level
- ajkjk 2y agoEr? No I'm describing actual physics. Physics does not care about rational vs reals. Nothing is measured to infinite accuracy anyway.
- afiori 2y agoOne thing that is used a lot in physics are known as Dirac Deltas[0] that is, in very informal terms, the derivative of the function f(x) = 0 for negative x otherwise f(x) = 1. Physics are very good at working with concepts and abstraction before any formal mathy justifications can be found, but the only way to formaly work with a dirac delta that makes sense formally is defining it in terms of measures [0] https://en.wikipedia.org/wiki/Dirac_delta_function https://en.wikipedia.org/wiki/Dirac_delta_function
- ajkjk 2y agoThat is not true, it is not the 'only way' to formally deal with them. A better way to think of them is as the vector space dual of functions (/forms) under the pairing given by integration. No measures required. The measure theoretic explanation is very much "fitting delta functions into our existing machinery" rather than any sort of inherent requirement. Actually an even better way to think of delta functions is just as a geometric object, a point (or line/plane/etc). Which is somewhat related to the measure theoretic version, but much more simple to think about.
- staunton 2y agoYou need measures to define the vector spaces that those (what you call "forms", and others call "distributions") act on... Not that it would be impossible to define everything without measures (or even without real numbers), but it's really not clear how to go about it.
- tsimionescu 2y agoEven if accuracy is finite, the fact that, for example, circles aren't polygons is definitely relevant to physics. You might be able to get all of the relevant physics you need without the continuum by working with several disjoint sets of numbers (the rationals, the rationals-multiplied-by-pi, the rationals-multiplied-by-e, etc) but I'm not even sure of that.