A question on continuity of probability functions

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Discussion Overview

The discussion revolves around the continuity of probability functions, specifically addressing a proof presented in Saeed Ghahramani's "Fundamentals of Probability." Participants explore the definitions and implications of continuity in the context of set functions, questioning the applicability of classical theorems of calculus to this scenario.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions how a theorem of continuity for functions from R to R can be applied to a set function from a collection of sets to R.
  • Another participant points out that the proof only addresses increasing and decreasing sequences, not general convergent sequences, which raises concerns about the completeness of the argument.
  • A later reply clarifies that the term "classic theorem" refers to the definition of continuity, but questions remain about the specific function being discussed.
  • There is a discussion about the nature of convergence, with one participant asserting that increasing or decreasing sequences can be convergent, while another insists that proving continuity for these sequences does not suffice for general convergence.
  • Some participants express confusion over terminology and request clearer definitions and questions to facilitate understanding.
  • One participant emphasizes the importance of distinguishing between different types of mathematical statements, such as theorems, definitions, and axioms.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the application of continuity theorems to probability functions. There are multiple competing views regarding the definitions and implications of continuity, convergence, and the nature of the mathematical statements being discussed.

Contextual Notes

Limitations in clarity and understanding are evident, with participants expressing confusion over terminology and the specific mathematical constructs being referenced. The discussion reflects a mix of technical reasoning and conceptual challenges without resolving the underlying issues.

Mathelogician
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Hi everybody! In Saeed Ghahramani's "fundamentals of probability" he proves the continuity of the probability function f:P(S) ->[0,1] as follows:

He Defines the notions of increasing and decreasing sequences of sets (here sets of events) and then defines infinite limits of such sequences (as infinite union and infinite intersection of the sets of events; respectively) and then he claims proving the continuity of f using a way the following classic theorem of calculus:
f : R -> R is continuous on R if and only if, for every convergent sequence {xn} n =1 to infinite in R, limf({xn})=f(lim{xn}) as n goes to infinity.
But
1-the classic theorem is for functions from R to R; how can he use it for a set function (which is from a collection of sets to R, here)
2- Even if he is right, he only proves the continuity of the cases of increasing and decreasing sequences; not convergent(which he has not defined!) in general.

Now what to do with this problem?!
Regards.
 
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Let me know if i am not clear enough!
Regards.
 
Speaking for myself, I have difficulty responding.
Your text looks more like statements than like questions.
Furthermore, I did not read your book and your statements are not clear enough for me.
It also seems like you are using math words not entirely correctly, making it difficult to understand your problem.

Let's take a look at some of your words.
You mention the classic theorem of calculus.
I know of no such thing, but reading on, it seems you meant the definition of continuity.
Did you?

Then you mention a set function (being a function on a collection of sets), but I do not understand which function you mean.
Can you clarify?

It may be that I can figure out what you mean if I look long enough and try to understand long enough.
But to be honest, I don't feel like spending that time.

Perhaps you can state your questions more like questions and with clear references?
 
Here you can find all the story:

Regards.
 
There does not seem to be a reference to a classic theorem of calculus.
But there is a reference to how continuity is defined according to calculus.

I don't see a function that is defined on a collection of sets.

What is your question exactly?
 
OK! as i mentioned at first, by classic i meant the same theorem of continuity (using convergent sequences)
The probability function IS defined on the collection of sets of events P(S) [Indeed the power set of the set S of sample space!] see line 8 in first page!

But the theorem of calculus is:
1- For functions from R to R .[Not any set like P(S) to R]
2- Used for convergent sequences [Not increasing or decreasing sequences!]

Now is that clear?!
 
Mathelogician said:
OK! as i mentioned at first, by classic i meant the same theorem of continuity (using convergent sequences)
The probability function IS defined on the collection of sets of events P(S) [Indeed the power set of the set S of sample space!] see line 8 in first page!

But the theorem of calculus is:
1- For functions from R to R .[Not any set like P(S) to R]
2- Used for convergent sequences [Not increasing or decreasing sequences!]

Now is that clear?!

I'll continue with asking for clarifications if you don't mind.
Usually it helps if we can get all symbols and definitions clear.

You mention R as opposed to R.
What do you mean by R? Is it different from the set of real numbers?

On an off note, a theorem is the same as a proposition, a lemma, a corollary, identity, rule, law, or principle.
They all follow from a given set of axioms (or postulates), but mathematically there is no clear distinction between them.
A theorem is different from an axiom or postulate.
It is also different from a definition.
If I understand you correctly, you're not talking about a theorem of continuity, but about its definition. See for instance here.

As for convergent sequences, there is no reason why an increasing or decreasing sequence wouldn't be convergent.
For instance 1/x is a decreasing sequence that converges to zero.
It is just an attribute of a convergent sequence whether it is increasing, decreasing, alternating, or something else.
 
R is the set of real numbers!

The theorem that i mentioned is can also be used as an equivalent definition of continuity![ If you want to ask what is the meaning of "equivalent", the i suggest you a first course in mathematical logic!] [[And the other definition of continuity is the one concluded by the equation(for continuity of f in the point a) Lim f(x) = f(a) as x->a ; and the epsilon-delta definition of limitation - and again if you don't know the epsilon-delta definition of limits, you are invited to review a first course on Calculus!]]

And about the convergence: convergence is not equivalent to increasing/decreasing. So if we are supposed to prove something about convergent sequences, we are not allowed to prove it for increasing/decreasing sequences![Or if so, we got to mention that it's for increasing/decreasing convergent sequences; and then we have to prove the assertion for non-increasing/non-decreasing cases!]

---------------
I think here in this topic, we have logic problem more than math!
 
Mathelogician said:
R is the set of real numbers!

The theorem that i mentioned is can also be used as an equivalent definition of continuity![ If you want to ask what is the meaning of "equivalent", the i suggest you a first course in mathematical logic!] [[And the other definition of continuity is the one concluded by the equation(for continuity of f in the point a) Lim f(x) = f(a) as x->a ; and the epsilon-delta definition of limitation - and again if you don't know the epsilon-delta definition of limits, you are invited to review a first course on Calculus!]]

And about the convergence: convergence is not equivalent to increasing/decreasing. So if we are supposed to prove something about convergent sequences, we are not allowed to prove it for increasing/decreasing sequences![Or if so, we got to mention that it's for increasing/decreasing convergent sequences; and then we have to prove the assertion for non-increasing/non-decreasing cases!]

---------------
I think here in this topic, we have logic problem more than math!

Your statements are correct, although your tone appears to be a bit disrespectful.
You do realize that this forum consists of volunteers?

I still don't understand your usage of R as opposed to R.
Nor do I understand which function on a collection of sets you intended.
And you seem to have misinterpreted my statement about convergence.

Anyway, it appears you have no questions.
 
  • #10
ILikeSerena said:
Anyway, it appears you have no questions.

Well said.

This thread is only heading to worse places so I'm going to close it.
 

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