Thread: want to show that two infinite summations are equal

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DESCRIPTION OF THE PROBLEM AND INFORMATION I AM SEEKING
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I would like to show that the following two infinite summations are equal for any two bijective functions f and g, which are defined from a set of natural numbers to a subset of real numbers. Both the set of natural numbers and the set of subset of reals are countably infinite. And the Bijection condition on these functions suggests that there exists a one to one correspondence between the elements of the set of natural numbers and the subset of reals.

Ok, so the expression I want to prove to be true is as follows:


sum (0 to infinity) [f(n) * prob {x| x = f(n)}]
=
sum (0 to infinity) [g(n) * prob {x| x = g(n)}]


In the above expression the function prob takes as input a set and returns a real value between 0 and 1. It is a

probability measure function that returns a real value corresponding to each of its set argument.

The two functions are bijective and are defined on the same two sets, that is they both map all elements of set of natural numbers to unique elements of a countably infinite subset of set of real numbers.

I am not quite sure how to proceed with the proof and need help.

This is what I know and why I think the two infinite summations should be equal.

Each summation is over the product of two terms. Both terms in the product are functions of n.

Both functions map the same set of natural numbers to the same subset of reals.
So if I build a set of all values of f(n) and another set that consists of all values of g(n), these sets will be equal.

The probability function just assigns a real value to its set argument.

Therefore, for all values of n which is a natural number, a set that consists of all possible real values of type ( f(n)*porb {x|x = f(n)} ) will be equal to the set that consists of all possible real values of type ( g(n)*prob {X|x = g(n)} ).

Lets call these sets A and B respectively.

Now since both sets A and B are countably infinite and are equal, the sum (0 to infinity) of all elements of these sets will also be equal.

I am not a mathematics major and am not quite sure how to proceed with the proof and need help.
I want to state above reasoning mathematically and as unambiguously as possible.

Can some one please comment on my reasoning and point out if there are any mistakes? May be suggest a better way

Both functions map the same set of natural numbers to the same subset of reals.

I think that is the key point (I wish it was nearer the top!). If i interpret you correctly, all you are really doing here is changing the order of summation, and you know that will not affect the value of the sum. (i think that is also the gist of what you were saying)


To state it rigourously, you could show that there is a 1-1 mapping between the terms in the first sum and the terms in the second sum.


Apologies if i have misunderstood your problem.

Dear Ormkarr

Thank you so much for your detailed response. Unfortunately I could not write back any sooner. The reply to my post seem to have disappeared from the forum. I was wondering if it can be restored. Your rediscription of the problem was accurate and very helpful for me, I am now better able to understand the problem and am learning how to clearly describe probelms.
I am sorry about the missing information. Here is some other information I know about the problem. The probability function in my problem is a probability measure such that it satisfies all the basic axioms of probability, that is, its value is between 0 and 1 both inclusive. The probability of empty set is 0 and that of the universe is 1. The probability of union of two disjoint sets is the sum of their individual probabilities, etc.

I guess the clue to the proof is that if a series is absolutely convergent then any possible rearrangement of it would sum to a unique value. I was wondering if you could help me and guide me to get a detailed proof for this problem.

Thanks in adavnce