- #1
Pikkugnome
- 10
- 2
Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?
Non-measurable sets are fairly pathological.Pikkugnome said:Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable.
Sets that are relevant to physical phenomena are generally measureable.Pikkugnome said:Is there a reason why the choice is also preferred in physics?
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:Non-measurable sets are fairly pathological.
Can you think of any non-measurable set that would be of interest in physics? I can't.PeroK said:Sets that are relevant to physical phenomena are generally measureable.
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:Can you think of any non-measurable set that would be of interest in physics? I can't.
PeroK said:Is the axiom of choice ever relevant to mathematical physics?
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:Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?
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:If Banach Tarski could be physically realizable, diamonds, gold, would be worthless.
I can just see Marilyn Monroe singing "Banach-Tarski is a girl's best friend"!WWGD said:If Banach Tarski could be physically realizable, diamonds, gold, would be worthless.
And Marylin Idiot Savant is Probability/Mathematics ' biggest enemy *PeroK said:I can just see Marilyn Monroe singing "Banach-Tarski is a girl's best friend"!
You wanted to write down their name, but the margin was too..fresh_42 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.
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: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.
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:There's also the magic on how that collection of discrete , finite, points magically turns into a continuum, with nonzero length, area, etc.
The same applies for all dimensions. And for 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.
The Lebesgue measure is a mathematical concept that is used to measure the size or volume of a set in n-dimensional space. It is named after French mathematician Henri Lebesgue and is a fundamental tool in the field of measure theory.
The Lebesgue measure differs from other measures, such as the Riemann or Borel measures, in that it is defined for a wider class of sets. It can measure sets that are not necessarily "nice" or well-behaved, such as non-measurable sets or sets with fractal-like properties.
In probability theory, the Lebesgue measure is used to define the probability of an event occurring. It allows for the calculation of probabilities for events that are not necessarily "simple" or easily definable. This makes it a powerful tool in the study of random variables and stochastic processes.
The Lebesgue measure is an integral part of basic probability theory as it is used to define the probability of an event occurring. It is also used in the calculation of expected values and in the formulation of important theorems, such as the Law of Large Numbers and the Central Limit Theorem.
Yes, the Lebesgue measure has many real-world applications, particularly in fields such as physics, economics, and engineering. It is used to measure the volume of irregularly shaped objects, to model the behavior of complex systems, and to analyze data in various industries. It is also used in the development of algorithms for data analysis and machine learning.