Sample spaces and topologies

[Mike2]

Both point set topologies and sample spaces work with unions and intersections of their elements. Point set topologies have distance functions between points, and sample spaces have probability density function at each element. Are there any texts or studies that combine these two disciplines or describe some features of one in terms of the other? Thanks.

I'm wondering if quantum mechanics might be a sample space description and general relativity might be a topological description of the same space.

 

[matt grime]

Both point set topologies and sample spaces work with unions and intersections of their elements. Point set topologies have distance functions between points,

no they don't. Only metrizable topological spaces may be equipped with a metric that is compatible with the topology. Very few topological spaces come with a natural metric.

I'm wondering if quantum mechanics might be a sample space description and general relativity might be a topological description of the same space.

You want to look at functional analysis.

 

[Mike2]

You want to look at functional analysis.

Do you have a recommendation on a well written book on the subject at say the beginning/intermediate level? Thanks.

 

[mathman]

Sample spaces are special cases of measure spaces, i.e. essentially they can be viewed as measure spaces with total measure=1. They don't need any topology.

On the other hand topological spaces don't necessarily have any measure associated with them.

 

[Mike2]

Sample spaces are special cases of measure spaces, i.e. essentially they can be viewed as measure spaces with total measure=1. They don't need any topology.

On the other hand topological spaces don't necessarily have any measure associated with them.

Yes, I know that. I was struck with the similarity of unions and intersections in both spaces. And I was wondering how artificial was the added struction to turn one into a sample space and the other into a metric space. The sample space seems to have a scalar field, a probability density function, (in the continuous case). The metric seems to be a 2-form field (in the continuous case). Are these added structures completely arbitrary, or is there some underlying logic to them when viewed in a particular way? Certainly in the sample space one can count the elements in an event and compare them to the total number of elements to come up with a probability - this seems like a natural thing to do. Can the same kind of thing be said of a metric space? For example, is a metric just another way of giving a number for a relationship between two elements/points - one is with respect to the whole, the other is with respect to another?

If metrics and probability density functions can naturally be imposed on any space of elements, then constraints in one system might inform the other which might even begin to appear physical. What do you think?

 

[mathman]

Yes, I know that. I was struck with the similarity of unions and intersections in both spaces.

Unions and intersections are operations on sets. Set theory does not require either topology or measure. These make use of added definitions. However, both start with set theory.

 

[Mike2]

Unions and intersections are operations on sets. Set theory does not require either topology or measure. These make use of added definitions. However, both start with set theory.

I feel like I'm starting to confuse myself with many issue, and I could use some clarification. Is there a mathematical entity defining the probability of two samples in a sample space occurring? Is this a conditional probability or joint probability? Is it always as simple as the probability of obtain the one sample times the probability of obtaining the second sample?

 

[HallsofIvy]

All you are really saying is that they are both based on sets. Just about any mathematics can be based on sets!

 

[mathman]

I feel like I'm starting to confuse myself with many issue, and I could use some clarification. Is there a mathematical entity defining the probability of two samples in a sample space occurring? Is this a conditional probability or joint probability? Is it always as simple as the probability of obtain the one sample times the probability of obtaining the second sample?

P(A&B)=P(A)P(B) when they are independent, otherwise P(B|A)P(A).