Generalize the sentence in $R^d$

Question

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 $\mathbb{R}^d$?

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 $(-\infty, t)$ ?

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

Answer 1

Try with $$ \phi(t) = m\left(E\bigcap\prod_{i=1}^d (-\infty, t_i)\right). $$