Does this explicit and bijective measurable function exist and satisfy my motivation?

Question

This is a follow up to this question. The only difference is $f$ is bijective and the range of $f$ has to be $[0,1]$.

Main Question:

Using the Lebesgue outer measure, does there exist an explicit and bijective function $f:[0,1]\to[0,1]$ such that:

1. the function $f$ is measurable in the sense of Caratheodory
2. the graph of $f$ is dense in $[0,1]\times[0,1]$
3. the range of $f$ is $[0,1]$
4. the collection of all subsets of $[0,1]$ (with pre-images under $f$ that are measurable in the sense of Caratheodory) is a non-perfect dense set in the collection of all subsets of $[0,1]$, where we define a topology with this answer:

Let ${\mathbb P}[0,1]$ be the collection of all subsets of $[0,1]$ modulo the equivalence relation $\sim$ defined by $E \sim F \Leftrightarrow {\lambda^{*}(E \Delta F)} = 0,$ where $\lambda^{*}$ is Lebesgue outer measure and $\Delta$ is the symmetric difference operation on sets. The set ${\mathbb P}[0,1]$ can be made into a complete metric space by defining the distance function $d,$ where $d(E,F) = {\lambda^{*} (E \Delta F)}.$

5. the graph of $f$ is non-uniform (i.e. without complete spacial randomness) in $[0,1]\times[0,1]$
6. using the Lebesgue measure, the expected value of $f$ is computable

Simplified Version of the Main Question:

In the main question, a non-perfect dense set in the collection of all subsets of the range of $f$ is “topologically large” in the collection of all subsets of the range of $f$, where the main question translates to the following:

Does there exist a explicit, bijective and measurable function $f:[0,1]\to[0,1]$ such that:

1. the graph of $f$ is dense in $[0,1]\times[0,1]$
2. the range of $f$ is $[0,1]$
3. the pre-image of most subsets of the range of $f$ under $f$ are measurable in the sense of Caratheodory
4. the graph of $f$ is non-uniform (i.e. without complete spacial randomness) in $[0,1]\times[0,1]$
5. using the uniform probability measure, the expected value of $f$ is computable?

Motivation: I wanted to define an explicit and bijective function $f:[0,1]\to[0,1]$ where the graph of $f$ is somewhat but not too evenly distributed (i.e. with complete spacial randomness) in $[0,1]\times[0,1]$, such that using the uniform probability measure, we want a subset $X\subseteq[0,1]$, where (when function $f:[0,1]\to[0,1]$ is restricted to $f:X\to[0,1]$) the expected value of $f$ is undefined so we can find an unique extension of the expected value of $f$ which gives a finite value instead.

Question on motivation: If the function from the main question exists, does it satisfy the motivation?

Attempt to Solve Both Questions:

I can't prove an explicit example exists but here is my attempt from this and this question (note in both links, neither answers gave what I wanted, e.g. in this answer, the function wasn't very explicit and in this answer, the function was extremely “non-uniform”):

In case one wants to read here, here's the attempt:

Suppose the base-$3$ expansion of real numbers, in interval $[0,1]$, have infinite decimals that approach $x\in[0,1]$ from the right side so when $0\le x_1,x_2\le 1$ (and $x_1=x_2$) we get $f(x_1)=f(x_2)$.

Furthermore, for $\mathbb{N}\cup\left\{0\right\}=\mathbb{N}_{0}$, if $r\in\mathbb{N}_{0}$ and $\text{digit}_{3}:\mathbb{R}\times \mathbb{Z}\to\left\{0,1,2\right\}$ is a function where $\text{digit}_{3}(x,r)$ takes the digit in the $3^{r}$-th decimal fraction of the base-$3$ expansion of $x$ (e.g. $\text{digit}_{3}(1.789,2)=\text{digit}_{3}({1.210022{\cdot\cdot\cdot}}_{3},2)=1$), then $\left\{{g_r}^{\prime}\right\}_{r\in\mathbb{N}_{0}}$ is a sequence of functions (and $\left[\cdot\right]$ is the nearest integer function) such that ${g_r}^{\prime}:\mathbb{N}_0\to\mathbb{N}_0$ is defined to be:

\begin{equation} g_r^{\prime}(x)=\left[\frac{10}{3}\sin(rx)+\frac{10}{3}\right] \end{equation}

then for some function $k:\mathbb{N}_{0}\to\mathbb{N}_{0}$, where $k$ is strictly increasing and $k(0)$ is a positive number, we want the the intermediate function (before $f$) or $f_{1}:[0,1]\to[0,10]$ to satisfy the main question (such that, in criteria 3. the range of $f_1$ is $[0,10]$).

\begin{alignat}{2} & f_{1}(x) = &&\left|\left(\sum\limits_{r=0}^{\infty} g_{r+1}^{\prime}\!\left(\sum\limits_{p=r}^{r+k(r)}\text{digit}_{3}(x,p)\right)\!\!\bigg/3^{r}\right)-10\right|= \label{eq:025} \\ & && \left|\left(\left(\sum\limits_{r=0}^{\infty}\left[\frac{10}{3}\sin\left(\left(r+1\right)\left(\sum\limits_{p=r}^{r+k(r)}\text{digit}_{3}(x,p)\right)\right)+\frac{10}{3}\right]\right)\!\!\bigg /3^{r}\right)-10\right| \nonumber \end{alignat}

(One example of $k(r)$ that may satisfy main question, when with criteria 3. the range of $f_1$ is $[0,10]$ instead of $[0,1]$, is $k(r)=10r+20$)

What we’re doing with $f_1$ is we are converting every digit of the base-$3$ expansion of $x$ into a pseudo-random number that is non-equally likely to be an integer, including and in-between, $0$ and $20/3$. Furthermore, we attempt to make the function dense in $[0,1]\times[0,10]$ by adding the $3^{r}$-th decimal fraction with the next $3^k$-th decimal fractions; however, we also want to control the end-points of $[0,10]$, such that $f_1$ is dense in $\left[0,1\right]\times\left[0,1\right]$ by manipulating $f_1$ to get:

\begin{alignat}{2} & f(x) = && 1-\frac{1}{10}f_1(x)\label{eq:109}\\ & && 1-\left(\frac{1}{10}\right)\left|\left(\left(\sum\limits_{r=0}^{\infty}\left[\frac{10}{3}\sin\left(\left(r+1\right)\left(\sum\limits_{p=r}^{r+k(r)}\text{digit}_{3}(x,p)\right)\right)+\frac{10}{3}\right]\right)\!\!\bigg/3^{r}\right)-10\right| \nonumber \end{alignat}

(e.g. where $k(r)=10r+20$). You can use programming to visualize $f$ though I don't know if you can graph the entire function. (The programming I used is mathematica.)

Clear["Global`*"]
k[r_] := k[r] = 
  20 (* You can adjust k[r]; however, mathematica is unable to graph \
f when k[r] is steepy increasing e.g. for this function, k[r] must be \
less than 25 for the code to show a graph. Instead, it should be k[r]=10r+20 *)
g1[xr_, r_] := 
 g1[xr, r] = 
  Round[(10/3) Sin[r xr] + (10/
      3)] (Converts the (3^r)th decimal fraction,in the base 3 

expansion of the x-values in[x1,x2] (defined as xr or x_r not xr) 

into a psuedo-random number that's non-equally likely to spit a 

number between,and including, 0 and 20/3 *)
f[x_] := f[x] = 
  N[1 - ((1)/(10)) RealAbs[
      Sum[g1[Sum[
           RealDigits[x, 3, k[r], -r][[1]][[z]], {z, r + 1, k[r]}], 
          r + 1]/(3^r), {r, 0, 8}] - 
       10]] (Defines function f,I assume the larger k[r]'s values, the more 

the function appears dense in [0,1]x[0,1])
p = .00005 (Incremement between the x-values in the points of the 

graph below)
ListPlot[Table[{x, f[x]}, {x, p, 1, 
   p}]] (Graphs countable points of the functions but is not a 

complete accurate graph. There are uncountably many points that need 

to be included.)

Unfortunately, I only studied up to intro to advanced mathematics. (Without a deep undestanding of math I'm unable to prove if the function gives what I'm looking for.)

Is there a simpler example?

Observation From This Post:

From this, it appears $f$ might satisfy criteria 4. of the main question if the pre-image of every subset of $[0,1]$ under $f$ has zero measure. (Note the subset of a measure zero set is measurable.) Since the graph of $f$ must be dense in $[0,1]×[0,1]$, the points in the graph of $f$ should be "spread out enough" in $[0,1]×[0,1]$, such that every pre-image of every subset of $[0,1]$ under $f$ has zero measure. (I don't know if we can explicitly define a simple example that allows this.)