Non-measurable sets are fairly pathological.Pikkugnome said:
Sets that are relevant to physical phenomena are generally measureable.Pikkugnome said:
Good point. But I wonder if they might actually be more numerous than measurable sets, like the transcendental numbers versus the algebraic numbers.PeroK said:
Can you think of any non-measurable set that would be of interest in physics? I can't.PeroK said:
I think the question was asked on here a few years ago, in a slightly different context. To obtain a subset of ##\mathbb R## that is not Borel-measurable requires the axiom of choice. Is the axiom of choice ever relevant to mathematical physics?FactChecker said:
PeroK said:
Because measurable sets have the properties we believe any physically measurable things should have. These are encoded in the properties of a sigma algebra. If you extended probabilities to non-measurable sets you would open the door to a whole set of paradoxes.Pikkugnome said:
I remember a colloquium about the proof of Banach-Tarski. The referent argued that it is less the AC that is against our intuition, rather it is our concept of a point that lacks any physical reality. This is an interesting point of view since it is primarily AC that is considered the culprit. But the more I think about it the more I have to agree to that professor whose name I have unfortunately forgotten.WWGD said:
I can just see Marilyn Monroe singing "Banach-Tarski is a girl's best friend"!WWGD said:
And Marylin Idiot Savant is Probability/Mathematics ' biggest enemy *PeroK said:
You wanted to write down their name, but the margin was too..fresh_42 said:
There's also the magic on how that collection of discrete , finite, points magically turns into a continuum, with nonzero length, area, etc.fresh_42 said:
The subject of this thread is somehow reached again. We ignore points since they have no positive Lebesgues measure. But without them, we wouldn't have science as we use it today. I don't think we would get very far with only three-dimensional objects.WWGD said:
The same applies for all dimensions. And for length/area, etc.fresh_42 said:
Lebesgue measure is a way of assigning a volume, area, or length to subsets of a given space, typically the real numbers or Euclidean space. In probability theory, it is often used to define the probability of events in a continuous probability space, providing a rigorous foundation for integration and measure theory.
Lebesgue measure generalizes the concept of "length" to more complex sets and is more flexible than the Riemann measure. While the Riemann measure only works well for simple sets like intervals, the Lebesgue measure can handle more complicated sets, including those that are not necessarily intervals or unions of intervals. This makes Lebesgue measure particularly useful in probability theory for defining integrals of more complex functions.
Lebesgue measure is crucial in probability theory because it allows for the proper definition of integrals for a wider class of functions, which is essential for calculating probabilities and expected values. It also supports the rigorous formulation of continuous probability distributions, such as the normal distribution, and provides a solid foundation for advanced topics like stochastic processes and measure-theoretic probability.
An example of a probability distribution defined using Lebesgue measure is the uniform distribution on an interval [a, b]. The probability density function (PDF) of this distribution is f(x) = 1/(b-a) for x in [a, b] and 0 otherwise. The Lebesgue measure ensures that the integral of the PDF over the interval [a, b] equals 1, which is a requirement for any probability distribution.
In probability theory, sets of measure zero are considered negligible or "almost impossible" events. The Lebesgue measure assigns a measure of zero to these sets, meaning they have no impact on the probability calculations. This concept is used to handle events that are theoretically possible but have no practical significance, ensuring that the probability measure remains consistent and well-defined.