Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$.

**Question:**

Does there exist an explicit and *bijective* $f:\mathbb{R}\to\mathbb{R}$, where:

- The function $f$ is measurable
- The range of $f$ is $\mathbb{R}$
- For all real $x_1,x_2,y_1,y_2$, where $-\infty<x_1<x_2<\infty$ and $-\infty<y_1<y_2<\infty$: $$\lambda(\left([x_1,x_2]\times[y_1,y_2]\right)\cap\left\{(x,f(x)):x\in\mathbb{R}\right\})>0$$

$\quad\quad\!$ and $$\lambda(\left([x_1,x_2]\times[y_1,y_2]\right)\cap\left\{(x,f(x)):x\in\mathbb{R}\right\})\neq(x_2-x_1)(y_2-y_1)$$

**Attempt:** I'm not sure how to answer this question but I heard of Conway’s Base-13 function?

I have also asked a similar question here with an answer suggesting that no function satisfies that question.

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