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 $f_4:\mathbb{Q}\cap[0,1]\to\left\{0,1\right\}$

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