How do we describe an intuitive arithmetic mean that gives the following? (I can't type more than 200 letters)

Question: How do we describe an intuitive arithmetic mean that gives a defined value between the infimum and supremum of a function's range (and coincides with other arithmetic means) without peice-wise notation for a maximum class of functions?

Background: I am an undergraduante and I stated this question several times but it comes out is as unclear. See what you interpret and solve it to the best of your ability, perhaps my own answer (which I will later show) gives clues.

Details:
Suppose we have $f:A\to B $  where $A\subseteq \mathbb{R}$ and we are given a class of measures $\mu_1,\mu_2,\mu_3,...\mu_{m}$ (each with their own sigma algebras) such that the arithmetic average of each measure is

$$\frac{1}{\mu_{i}(A)}\int_{A}f(x)d\mu_{i}$$

Could we describe an intuitive arithmetic average from measure $\mathcal{M}$ such that it:

1. Coincides other arithmetic averages of other specific functions where their averages are defined

For example, if $\mu_{1}$ is the Lebesgue Measure, then $\mathcal{M}$ should give the same average as that from the Lebesgue Measure when $\mu_1(A)$ is positive since the average from the Lebesgue Measure is defined for that case.

If  $\mu_{m}$ is the counting measure then $\mathcal{M}$ should give the same average as $\mu_{m}$ when $A$ is finite and contains more than one element since the average from the counting measure is defined for that case.

While the solutions seems to be Haar Measure; and, when $A$ has zero Lebesgue Measure and infinite points there exists dimension $d\in[0,1]$ where $s<d$ give a measure of $\infty$ and $s>d$ give a measure of $0$, if $d=s$ the Haursdorff Measure would be any number between zero and infinity and is therefore not unique.

2.  Could we describe an intuitive average that follows 1. and has the smallest number of rows in a piece-wise array which defines the arithmetic average from measure $\mathcal{M}$

3. Could we find an average that follows 1. and 2. and is defined between the infimum and supremum of the range when the average from $\mathcal{M}$ is defined. 

4. Could we find an average that first follows the previous three rules and is defined for a maximum class of functions.


Lastly consider these examples. How do we apply our average from $\mathcal{M}$ to these examples:

Example 1:

Consider function $f_1(x)$ where the first sub-domain is:

$$S_1=\left\{\frac{1}{\sqrt{2s^2}}:s\in\mathbb{N}\right\}$$ And the second domain is: 

$$S_2=\left\{\frac{1}{2^s}+\frac{1}{2^t}:s,t\in\mathbb{N}\right\}$$

Then $f_1:S_1\cup S_2\to\left\{0,1 \right\}$ such that

$$ f_1(x)=\begin{cases} 1 & x\in S_1 \\ 0 & x\in S_2 \end{cases} $$

Second Example:

Consider function $f_2$ defined as the following: Suppose $G_{0}=[0,1]$ and for $n\in\mathbb{Z}^{+}$. $$G_{n+1}=\frac{G_{n}}{5}\cup\frac{G_{n}+2}{5}\cup\frac{G_{n}+4}{5}$$

Hence the first sub-domain $\mathcal{G}$ is defined as:

$$\mathcal{G}=\left\{\bigcap\limits_{n=1}^{\infty}G_n:G_0=[0,1],G_n=\frac{G_{n-1}}{5}\cup\frac{G_{n-1}+2}{5}\cup\frac{G_{n-1}+4}{5}\right\}$$

The second sub-domain is the Cantor Set (defined as $\mathcal{C}$) and the third sub-domain is the rational numbers (defined as $\mathbb{Q}\cap[0,1]$). 

Therefore, we define $f_2:\left(\mathcal{C}\cup\mathcal G\cup\mathbb{Q}\right)\cap[0,1]\to\left\{1,2,3\right\}$

$$f_2(x)=\begin{cases} 3 & x\in\mathcal{G}\setminus\mathbb{Q} \\ 2 & x\in\mathcal{C}\setminus\left(\mathcal{G}\cup\mathbb{Q}\right) \\ 1 & x\in\mathbb{Q}\cap[0,1] \end{cases}$$

Third Example:

Suppose we want to define $f_3$ such that if $\mathcal{C}$ is the Cantor Set, the first sub-domain of $f_3$ is:

$$S_n=\bigcup\limits_{n=0}^{\infty}\bigcup\limits_{k=0}^{2^n-1}\left\{3^{-n}(x+k):x\in\mathcal{C}\right\}$$

If $\mathcal{G}$ is the first sub-domain in second example (section 0.2), the second sub-domain of $f_3$ is:

$$T_n=\left(\bigcup\limits_{n=0}^{\infty}\bigcup\limits_{k=0}^{3^n-1}\left\{5^{-n}(x+k):x\in\mathcal{G}\right\}\right)\setminus S_n$$

Therefore $f_3:S_n \cup T_n\to\{1,2\}$

$$f_3(x)=\begin{cases} 2 & x\in S_n \\ 1 & x\in T_n\\ \end{cases} $$

Example 4: 

Consider function $f_4(x)$ where $f_4:\mathbb{Q}\cap[0,1]\to\left\{0,1\right\}$

$$ f_4(x)=\begin{cases} 1 & x\in\left(\mathbb{Q}\setminus\left\{\frac{s}{2t+1}:s,t\in\mathbb{N}\right\}\right)\cap[0,1] \\ 0 & x\in\left\{\frac{s}{2t+1}:s,t\in\mathbb{N}\right\}\cap[0,1] \end{cases} $$

  • Just to make sure that I am understanding your questions clearly, you have a measurable space $\mathcal{M}$ and you would like to define the average of a measurable function f on M so that it pretty much works for all sorts of discrete and continuous measures?

  • Yes, I want an average that works for discrete and continuous measures and gives an average defined on the fewest rows of a piece-wise array.

  • Hey, Phillip I forgot to extend the due date. I gave plenty of time.

  • Just to clarify, you have a measurable space, and a given function f, and you are looking for a measure that gives the average of f over every measurable set? Or are you looking for a measure that works for every measurable function?

  • Although the average of f over A is not measure w.r.t. A due to the division involved.

  • I read the details of the attached file. But the question is still vague. I am going to write up more precise version(s) of it, and then proceed to answer it.

  • Martin, I believe I’m looking for a measure that gives the average of f over every measurable set. The thing is a measure such as Dirac mass can measure any set but it doesn’t coincide with the Lebesgue measure and counting measure.

  • As I wrote the average of f over A is not a measure, but we can view it as a function on the sigma algebra of measurable sets. This function is defined using the usual formula for sets with positive Lebesgue measure. And the question is how we can extend it to the family of all measurable sets. Of course a function can be trivially extended by assigning some arbitrary values to elements outside its domain. But we are looking for an extension of the notion of average that preserves some propertie

  • So the question becomes: what are the properties that you wish the average preserves? (Using different criteria we will probably get different answers, and even with a given set of properties the answer is probably not unique.)

  • Add the most properties that will give us a fewest remaining averages then we can evaluate how useful each one is. If possible, apply your favorite average to one of my functions. If you want I will pay you extra.

  • Actually, is it possible to test your average on all my functions. I wish I could calculate them myself but I have to measure and number theory.

  • Did you understand measure P stated in the paper? Sorry for the writing, but I was hoping someone could understand my answer.

  • Let me write up some clarifications about the problem and what I have got so far, and also compute the average of the example functions. Then we proceed from there.

  • Yes I have seen your definition of P, but I am not sure if there is a canonical way of constructing it. Or even if it exists for an arbitrary A.

  • P doesn’t exist for arbitrary A. It’s inner and outer measure form a sigma algebra that is slightly larger than the sigma algebra of the Lebesgue Measure. For sets that are not measurable by P, I added Density D but have a hard time defining it.

  • Hopefully your average coincides with the average from measure P and Density D.

  • I think standard deviation is crucial to finding an intuitive average see section 3.7 and 3.8 of https://documentcloud.adobe.com/link/review?uri=urn:aaid:scds:US:096cae63-8d0e-4183-9030-17ce69f21b92

  • Here's the link again: https://documentcloud.adobe.com/link/review?uri=urn:aaid:scds:US:096cae63-8d0e-4183-9030-17ce69f21b92 (do not include the time it was posted).

  • Sorry for yet another comment but suppose we want an average for functions defined on sets with an algebraic form, for example the rationals have the algebraic form m/n. Where m and n are integers.

  • No problem. I am working on the answer. It should be ready in a few days.

  • Martin, there’s three days left. How’s the progress going? Should I extend the due date?

Answer

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  • Let me know if any part is unclear.

  • Regarding the other question, working on this one took a quite long time. I did not keep the exact time, but writing and researching the answer took well above 10 hours, and was not really proportionate to the price. So for the other question, given its open-endedness and more vague nature, it should be priced higher than this one to justify the required time.

  • Martin thank you for you answer. You don’t have to clarify further. I have to improve my understanding of mathematics but I believe P isn’t simply the Lebesgue measure of S divided by the Lebesgue Measure of A. Measure P also is defined for sets where A has zero Lebesgue measure.

  • If I give you 500 for the next question is possible to improve P (and turn it into a measure).

  • I know what you have in mind about P. But as I explained in the file it is not possible to make P a measure and require it to assign desirable values at the same time. However if we do not require P to be a measure then the process that I described essentially does what we expect from P.

  • I accepted the other question. We can try to do (some of) the things you wanted to do with P using the family of generalized Hausdorff measures. I suggest you read the answer to this question with all its details, and then let me know what you want to further achieve with these tools in the other question.

  • I need to improve the definition of P. I tried explaining it on math stack exchange but no one could "accept" the definitions. I have two parts, one is P and the other is the density. (I hope to apply the definition of P and its average to my examples.)

The answer is accepted.