MHB How Can We Generalize Lebesgue Measurable Functions in Higher Dimensions?

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mathmari
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Hey! :o

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)
 
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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 $i=1,\dots,d$.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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