How do we define a unique, satisfying expected value from chosen sequences of bounded functions converging to an explicit, everywhere surjective function?

Consider the following attatchment [1]. See if there are research papers which address the topics in the attatched article. 

In the paper, consider the following:

Question 1: How do we define "satisfying" in $\S$3.2, so that $\mathbb{E}[f_r]$ answers $\S$1? 

Question 2: If question 1 is unclear, how do we improve $\S$4.2, so that the answer to $\S$4.2 answers $\S$3.2 and $\mathbb{E}[f_r]=\mathbb{E}[f_r^{\star}]$ answers $\S$1?

Question 3: What am I "measuring" in $\S$5.2? Is there a credible well-known version of my "measure"?

In case you need help, you can post concrete questions on Mathematics and Math Overflow. 

Finally, how much money should I offer?

Edit: I asked the support team to delete all the other files.

Edit 2: There were mistakes with the notations in $\S$3.1 onward. I have made adjustments.