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A question on continuity of probability functions

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Mathelogician

Member
Aug 6, 2012
35
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.
 

Mathelogician

Member
Aug 6, 2012
35
Let me know if i am not clear enough!
Regards.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
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?
 

Mathelogician

Member
Aug 6, 2012
35
Here you can find all the story:

Regards.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
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?
 

Mathelogician

Member
Aug 6, 2012
35
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?!!
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
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.
 

Mathelogician

Member
Aug 6, 2012
35
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!!]

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

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
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!!]

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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.
 

Jameson

Administrator
Staff member
Jan 26, 2012
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