Does the following function exist?

15 May 2023 0 2K Report

Does there exist a function f:[0,1]->[0,1] such

the function f is measurable in the sense of caratheodory

the graph of f is dense in [0,1] x [0,1]

the collection of all subsets of the range of f with pre-images under f that are measurable in the sense of Caratheodory is a non-perfect dense set (see https://math.stackexchange.com/questions/252448/probability-of-selecting-a-non-measurable-set/266090#266090) in the collection of all subsets of the range of f

f is non-uniform, i.e. without complete spacial randomness (see https://en.m.wikipedia.org/wiki/Complete_spatial_randomness#:~:text=Complete%20spatial%20randomness%20(CSR)%20describes,points%20within%20the%20defined%20area.) in [0,1] x [0,1]

using the uniform probability measure (see https://golem.ph.utexas.edu/category/2020/11/the_uniform_measure.html), the expected value of f is undefined?

Moreover, see the post in the link below for more info:

https://math.stackexchange.com/questions/4699353/does-the-function-in-the-description-exist

Note I need an explicit example so, using the uniform probability measure, the expected value of the function is undefined. This then requires the application of my paper in the attatchment. (See section 3.4 criteria (1) and section 4.1).

**Edit**: In 3. I converted non-measurable to measurable.

**Edit 2:** Changed 3. inorder for the question to make sense.

**Edit 3:** Last attempt to fix 3.

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