For all question related to the decreasing rearrangements of measurable functions so for measurable set too. As well the Lorentz spaces are strongly include since there are exclusively defined through the decreasing rearrangement of function. The analogue of this notion higher dimension is: The spherical (or symmetric, or radial) decreasing rearrangement of a measurable function
Let $f$ be a measurable function on a measurable space $(X,\mu)$. Then the decreasing rearrangement function of $f$ is the function $f^*$ defined on $[0,\infty)$ by
$$f^*(t) =\inf\lbrace s>0:\mu\lbrace x\in X: |f(x)|>s\rbrace \le t.\rbrace$$ This definition is similar to this used the following definition: Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. For a measurable set A, the rearrangement of A denote $A^* =B(0,r)$ is the ball centered at 0 with the same measure as $A$.
The spherical(or symmetric, or radial) decreasing rearrangement of a measurable function $f:\mathbb{R}^n \to \mathbb{R}$ is then defined by
$$f^*(x):=\int_0^{\infty} \chi_{\{|f|>t\}^*}(x)dt,$$
by comparison to the "layercake" representation of $f$, namely $$f(x)=\int_0^{\infty} \chi_{\{f>t\}}(x)dt.$$
