How Can We Generalize Lebesgue Measurable Functions in Higher Dimensions?

[mathmari]

Hey!

If $E \subset \mathbb{R}$ is Lebesgue measurable and $\phi(t)=m \left ((-\infty, t) \cap E\right )$, then $\phi$ is Lipschitz.

How could we generalize this sentence in $\mathbb{R}^d$?? (Wondering)

If $E \subset \mathbb{R}^d$ is Lebesgue measurable and $\phi(t)=m \left (\dots \cap E\right )$, then $\phi$ is Lipschitz.

What should be instead of $(-\infty, t)$ ?? (Wondering)

Maybe a rectangle in $\mathbb{R}^d$?? Or something else?? (Wondering)

 

[Jhoneine]

Yes, you can use a rectangle in $\mathbb{R}^d$. Specifically, the rectangle should be of the form $(-\infty, t_1)\times \cdots \times (-\infty, t_d)$. This is the set of all points $(x_1,\dots,x_d)$ such that $x_i < t_i$ for all $i=1,\dots,d$.